Harmonic Analysis and Partial Differential Equations: In Honor of Vladimir Maz'ya 3031254236, 9783031254239

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Anatoly Golberg Peter Kuchment David Shoikhet Editors

Harmonic Analysis and Partial Differential Equations In Honor of Vladimir Maz’ya

Anatoly Golberg • Peter Kuchment • David Shoikhet Editors

Harmonic Analysis and Partial Differential Equations In Honor of Vladimir Maz’ya

Previously published in Analysis and Mathematical Physics “Special Issue: Harmonic Analysis and Partial Differential Equations” Volume 10, issue 4, 2020, Volume 11, issue 1–4, 2021 and Volume 12, issue 2, 2022

Editors Anatoly Golberg Department of Mathematics Holon Institute of Technology Holon, Israel

Peter Kuchment Mathematics Department Texas A&M University College Station, TX, USA

David Shoikhet Department of Mathematics Holon Institute of Technology Holon, Israel

Spinoff from journal: “Analysis and Mathematical Physics” Volume 10, issue 4, 2020, Volume 11, issue 1–4, 2021 and Volume 12, issue 2, 2022 ISBN 978-3-031-25423-9 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Contents

Preface .................................................................................................................... ix The scientific work of Vladimir Maz’ya ................................................................ 1 A. Cialdea: Analysis and Mathematical Physics 2020, 2021:143 (27, July 2021) https://doi.org/10.1007/s13324-020-00421-z Topologicalmappings of finite area distortion .................................................... 15 E. Afanas’eva and A. Golberg: Analysis and Mathematical Physics 2022, 2022:54 (18, March 2022) https://doi.org/10.1007/s13324-022-00666-w On fractional Orlicz–Sobolev spaces.................................................................... 45 A. Alberico, A. Cianchi, L. Pick, and L. Slavíková: Analysis and Mathematical Physics 2021, 2021:84 (23, March 2021) https://doi.org/10.1007/s13324-021-00511-6 Interpolative gap bounds for nonautonomous integrals .................................... 67 C. De Filippis and G. Mingione: Analysis and Mathematical Physics 2021, 2021:117 (1, June 2021) https://doi.org/10.1007/s13324-021-00534-z Positive Liouville theorem and asymptotic behaviour for (p, A)-Laplacian type elliptic equations with Fuchsian potentials in Morrey space .................................................................................. 107 R. Kr. Giri and Y. Pinchover: Analysis and Mathematical Physics 2020, 2020:67 (30, October 2020) https://doi.org/10.1007/s13324-020-00418-8 Space quasiconformal composition operators with applications to Neumann eigenvalues ...................................................................................... 141 V. Gol’dshtein, R. Hurri-Syrjänen, V. Pchelintsev, and A. Ukhlov: Analysis and Mathematical Physics 2020, 2020:78 (9, November 2020) https://doi.org/10.1007/s13324-020-00420-0 Teichmüller spaces and coefficient problems for univalent holomorphic functions ......................................................................................... 161 S. L. Krushkal: Analysis and Mathematical Physics 2020, 2020:51 (6, October 2020) https://doi.org/10.1007/s13324-020-00395-y A review of some new results in the theory of linear elliptic equations with drift in Ld .................................................................................... 181 N. V. Krylov: Analysis and Mathematical Physics 2021, 2021:73 (8, March 2021) https://doi.org/10.1007/s13324-021-00508-1 v

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Fast computation of elastic and hydrodynamic potentials using approximate approximations .............................................................................. 195 F. Lanzara, V. Maz’ya, and G. Schmidt: Analysis and Mathematical Physics 2020, 2020:81 (10, November 2020) https://doi.org/10.1007/s13324-020-00400-4 Spectral properties of the logarithmic Laplacian ............................................. 215 A. Laptev and T. Weth: Analysis and Mathematical Physics 2021, 2021:133 (29, June 2021) https://doi.org/10.1007/s13324-021-00527-y L1 convergence of Fourier transforms ............................................................... 239 E. Liflyand: Analysis and Mathematical Physics 2021, 2021:91 (5, April 2021) https://doi.org/10.1007/s13324-021-00530-3 Failure of Fredholm solvability for the Dirichlet problem corresponding to weakly elliptic systems ........................................................... 251 D. Mitrea, I. Mitrea, and M. Mitrea: Analysis and Mathematical Physics 2021, 2021:85 (25, March 2021) https://doi.org/10.1007/s13324-021-00521-4 Local regularity of axisymmetric solutions to the Navier–Stokes equations............................................................................................................... 275 G. Seregin: Analysis and Mathematical Physics 2020, 2020:46 (10, September 2020) https://doi.org/10.1007/s13324-020-00392-1 Nonlinear resolvent and rigidity of holomorphicmappings ............................. 295 D. Shoikhet: Analysis and Mathematical Physics 2021, 2021:36 (7, January 2021) https://doi.org/10.1007/s13324-020-00463-3 “Smooth rigidity” and Remez-type inequalities................................................ 305 Y. Yomdin: Analysis and Mathematical Physics 2021, 2021:89 (2, April 2021) https://doi.org/10.1007/s13324-021-00516-1

Preface Anatoly Golberg1, Peter Kuchment2, David Shoikhet1 1

Holon, Israel College Station, TX, USA

2

Vladimir Maz’ya is a world leading analyst who has made major contributions in an astonishing variety of areas of partial differential equations, function theory, and harmonic analysis. An incomplete list of these includes isoperimetric and integral inequalities; spectral theory of the Schrödinger operator; theory of capacity and nonlinear potentials; boundary behavior of solutions of elliptic equations (especially with irregular boundaries); counterexamples for higher order version of Hilbert’s 19th problem; pseudodifferential operators; water waves; Sobolev spaces theory; singularities of solutions of non-linear PDEs; theory of multipliers between functional spaces; maximum principle for elliptic and parabolic systems; point-wise interpolation inequalities for derivatives; oblique derivative problems; ill-posed boundary value problems; elasticity theory; differential equations with operator coefficients; geometric analysis; and much more. His (eternally incomplete) list of publications contains almost 600 research papers and monographs. We are reluctant to give an exact figure, since it would clearly be an underestimate of the constantly growing number. He has published 26 monographs in English, while including the Russian and German versions, the number would grow to be around 40. One can find the list in the next section. Not knowing that Vladimir Maz’ya is a real person, one could be justified imagining that this is a pen name for a strong group of mathematicians, or even a whole institution. We, however, can guarantee to the readers that he is indeed a single (albeit outstanding) human being. Since such mathematical productivity is clearly insufficient for him, he wrote two books on the history of mathematics—one autobiographic and the other a treatise on the scientific life of J. Hadamard (for the latter, he along with his wife and co-author T. Shaposhnikova received a prize of the French Academy). Well, this was still not good enough, so Vladimir started writing (initially for his grandchildren) and publishing books of children tales, of which he has published four (in Russian). It is not surprising that Vladimir Maz’ya has received numerous honors, including -

1962—the inaugural “Young Mathematician” prize of the Le\-nin\-grad Mathematical Society 1990—Honorary Doctorate from Rostock University 1999—the Humboldt Prize 2000—Member of the Royal Society of Edinburgh 2002—Member of the Swedish Academy of Science 2003—jointly with Tatyana Shaposhnikova, the Verdaguer Prize of the French Academy of Sciences (for the scientific biography of J. Hadamard) ix

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-

2004—Celsius Gold Medal of the Royal Society of Sciences in Uppsala 2009—Senior Whitehead Prize by the London Mathematical Society 2012—inaugural class Fellow of the American Mathematical Society 2013—Member of the Georgian National Academy of Sciences

Professor Maz’ya has been cited on Google Scholar more than 27,000 times and has Hirsch citation index 64. Many international conferences have been organized in his honor, including the one this book reflects on (as the 12th book in his honor). One can find further scientific and personal information about Vladimir (including the list of other volumes dedicated to him) on his home page https://users.mai.liu.se/vlama82/ and from the following papers and references therein: -

-

Alberto Caldea, The scientific work of Vladimir Maz’ya, in this volume. M. S. Agranovich, Yu.D. Burago, B. R. Vainberg, M. I. Vishik, S.G. Gindikin, V.A. Kondrat’ev, V. P. Maslov, S.V. Poborchii, Yu.G. Reshetnyak, V. P. Khavin, M.A. Shubin, Vladimir Gilelevich Maz’ya (on his 70th birthday), Russian Math. Surveys, 63:1 (2008), 189–196. The introductory articles from the volumes mentioned above.

The Holon Institute of Technology (HIT), together with the Gelbart Research Institute for the Mathematical Sciences of Bar-Ilan University, and in association with RUDN—Peoples’ Friendship University of Russia and the International Society for Analysis, its Applications and Computation (ISAAC), have organized an International Conference “Harmonic Analysis and PDE” at HIT, Holon, May 26–31, 2019 dedicated to the 80th Birthday of Professor Maz’ya. This volume contains works of some of the participants of the conference. The complete list of participants is provided below.

List of Participants Dov Aharonov, Israel; Mark Agranovsky, Israel; Darya Apushkinskaya, Germany; Elza Bakhtigareeva, Russia; Akram Begmatov, Uzbekistan; Matania Ben-Artzi, Israel; Lucian Beznea, Romania; Haim Brezis, Israel; Daoud Bshouty, Israel; Viktor Burenkov, Russia; Alberto Cialdea, Italy; Michael Cwikel, Israel; Evgeny Derevtsov, Russia; Mark Elin, Israel; Aviv Gibali, Israel; Victor Gichev, Russia; Anatoly Golberg, Israel; Mikhail Goldman, Russia; Vladimir Gol’dshtein, Israel; Vladimir Golubyatnikov, Russia; Alex Iosevich, USA; Alexander Isaev, Australia; Lavi Karp, Israel; Aben Khvoles, Israel; Pekka Koskela, Finland; Gady Kozma, Israel; Gershon Kresin, Israel; Samuel Krushkal, Israel; Peter Kuchment, USA; Massimo Lanza de Cristoforis, Italy; Flavia Lanzara, Italy; Mark Lawrence, Kazakhstan; Nir Lev, Israel; Elijah Liflyand, Israel; Maria Elena Luna-Elizarrarás, Israel; Vladimir Maz’ya, Sweden; Giuseppe Mingione, Italy; Ron Peled, Israel; Yehuda Pinchover, Israel; Leonid Rossovskii, Russia; Vladimir Rovensky, Israel; Koby Rubinstein, Israel; Gregory Seregin, Russia; Armen Sergeev, Russia; Michael Shapiro, Israel; Tatyana Shaposhnikova, Sweden; David Shoikhet, Israel; Pavel Shvartsman, Israel; Alexander Skubachevskii, Russia; Mikhail Sodin, Israel; Boris Solomyak, Israel; Igor Verbitsky, USA; Alexander Ukhlov, Israel; Eduard Yakubov, Israel; Yosef Yomdin, Israel; Lawrence Zalcman, Israel; Leonid Zelenko, Israel

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We wish to our dear friend and colleague Vladimir Maz’ya many more years of healthy, happy, and productive life. We expect many new discoveries from him. Anatoly Golberg, Peter Kuchment, David Shoikhet

Fig. 1 Mathematics Books by Vladimir Maz’ya

The list below includes only English versions. Another dozen of Russian and German language texts are not listed (these can be seen in Fig. 1). 1. Burago, Yu. D.; Maz’ya, V. G., Potential theory and function theory for irregular regions. Translated from Russian Seminars in Mathematics, V. A. Steklov Mathematical Institute, Leningrad, Vol. 3 Consultants Bureau, New York 1969 vii+68 pp. 2. Cialdea, Alberto; Maz’ya, Vladimir. Semi-bounded differential operators, contractive semigroups and beyond. Operator Theory: Advances and Applications, 243. Birkhäuser/Springer, Cham, 2014. xiv+252 pp. ISBN: 978-3-319-04557-3; 9783-319-04558-0 3. Gel’man, Igor V.; Maz’ya, Vladimir G. Estimates for differential operators in half-space. Translated from the 1981 German original by Darya Apushkinskaya. EMS Tracts in Mathematics, 31. Euro\-pean Mathematical Society (EMS), Zurich, 2019. xvi+246 pp. ISBN: 978-3-03719-191-0 4. Kozlov, Vladimir; Maz’ya, Vladimir. Theory of a higher-order Sturm-Liouville equation. Lecture Notes in Mathematics, 1659. Springer-Verlag, Berlin, 1997. xii+140 pp. ISBN: 3-540-63065-1

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5. Kozlov, Vladimir ; Maz’ya, Vladimir. Differential equations with operator coefficients with applications to boundary value problems for partial differential equations. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 1999. xx+442 pp. ISBN: 3-540-65119-5 6. Kozlov, Vladimir; Maz’ya, Vladimir; Movchan, Alexander. Asymptotic analysis of fields in multi-structures. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1999. xvi+282 pp. ISBN: 0-19-851495-6 7. Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., Spectral problems associated with corner singularities of solutions to elliptic equations. Mathematical Surveys and Monographs, 85. American Mathematical Society, Providence, RI, 2001. x+436 pp. ISBN: 0-8218-2727-8 8. Kozlov, V. A.; Maz’ya, V. G.; Rossmann, J., Elliptic boundary value problems in domains with point singularities. Mathematical Surveys and Monographs, 52. American Mathematical Society, Providence, RI, 1997. x+414 pp. ISBN: 0-82180754-4 9. Kresin, Gershon; Maz’ya, Vladimir. Maximum principles and sharp constants for solutions of elliptic and parabolic systems. Mathematical Surveys and Monographs, 183. American Mathematical Society, Providence, RI, 2012. viii+317 pp. ISBN: 978-0-8218-8981-7 10. Kresin, Gershon; Maz’ya, Vladimir. Sharp real-part theorems. A unified approach. Translated from the Russian and edited by T. Shaposhnikova. Lecture Notes in Mathematics, 1903. Springer, Berlin, 2007. xvi+140 pp. ISBN: 978-3540-69573-8; 3-540-69573-7 11. Kuznetsov, N.; Maz’ya, V.; Vainberg, B., Linear water waves. A mathematical approach. Cambridge University Press, Cambridge, 2002. xviii+513 pp. ISBN: 0-521-80853-7 12. Maz’ja, Vladimir G., Sobolev spaces. Translated from the Russian by T. O. Shaposhnikova. Springer Series in Soviet Mathematics. Springer-Verlag, Berlin, 1985. xix+486 pp. ISBN: 3-540-13589-8 13. Maz’ya, Vladimir. Sobolev spaces with applications to elliptic partial differential equations. Second, revised and augmented edition. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 342. Springer, Heidelberg, 2011. xxviii+866 pp. ISBN: 978-3-642-15563-5 14. Maz’ya, Vladimir. Differential equations of my younger years. Translated from the Russian by Arkady Alexeev. With a fore\-word by V.P. Havin. Birkhäuser/ Springer, Cham, 2014. xiv+191 pp. ISBN: 978-3-319-01808-9; 978-3-319-01809-6 15. Maz’ya, Vladimir G., Boundary behavior of solutions to elliptic equations in general domains. EMS Tracts in Mathematics, 30. European Mathematical Society (EMS), Zürich, 2018. x+430 pp. ISBN: 978-3-03719-190-3 16. Maz’ya, Vladimir; Morozov, N.F.; Plamenevskii B.A.; Stupyali L., Elliptic boundary value problems. American Mathematical Society Translations--Series 2, v. 123, ISSN: 0065-9290, ISBN: 0-8218-3082-1, 1984, 268 pp. 17. Maz’ya, Vladimir; Movchan, Alexander; Nieves, Michael. Green’s kernels and meso-scale approximations in perforated domains. Lecture Notes in Mathematics, 2077. Springer, Heidelberg, 2013. xviii+258 pp. ISBN: 978-3-319-003566; 978-3-319-00357-3

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18. Maz’ya, Vladimir; Nazarov, Serguei; Plamenevskij, Boris. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. I. Translated from the German by Georg Heinig and Christian Posthoff. Operator Theory: Advances and Applications, 111. Birkhäuser Verlag, Basel, 2000. xxiv+435 pp. ISBN: 3-7643-6397-5 19. Maz’ya, Vladimir; Nazarov, Serguei; Plamenevskij, Boris. Asymptotic theory of elliptic boundary value problems in singularly perturbed domains. Vol. II. Translated from the German by Plamenevskij. Operator Theory: Advances and Applications, 112. Birkhäuser Verlag, Basel, 2000. xxiv+323 pp. ISBN: 3-76436398-3 20. Maz’ya, Vladimir G.; Poborchi, Sergei V., Differentiable functions on bad domains. World Scientific Publishing Co., Inc., River Edge, NJ, 1997. xx+481 pp. ISBN: 981-02-2767-1 21. Maz’ya, Vladimir; Rossmann, Jürgen. Elliptic equations in polyhedral domains. Mathematical Surveys and Monographs, 162. American Mathematical Society, Providence, RI, 2010. viii+608 pp. ISBN: 978-0-8218-4983-5 22. Maz’ya, Vladimir; Schmidt, Gunther. Approximate approximations}. Mathematical Surveys and Monographs, 141. American Mathematical Society, Providence, RI, 2007. xiv+349 pp. ISBN: 978-0-8218-4203-4 23. Maz’ya, Vladimir; Shaposhnikova, Tatyana. Jacques Hadamard, a universal mathematician. History of Mathematics, 14. American Mathematical Society, Providence, RI; London Mathematical Society, London, 1998. xxviii+574 pp. ISBN: 0-8218-0841-9 24. Maz’ya, V. G.; Shaposhnikova, T. O. Theory of multipliers in spaces of differentiable functions. Monographs and Studies in Mathematics, 23. Pitman (Advanced Publishing Program), Bos\-ton, MA, 1985. xiii+344 pp. ISBN: 0-273-08638-3 25. Maz’ya, Vladimir G.; Shaposhnikova, Tatyana O. Theory of Sobolev multipliers. With applications to differential and integral operators. Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 337. Springer-Verlag, Berlin, 2009. xiv+609 pp. ISBN: 978-3-54069490-8 26. Maz’ya, Vladimir G.; Soloviev, Alexander A., Boundary integral equations on contours with peaks}. Translated into English and edited by Tatyana Shaposhnikova. Operator Theory: Advances and Applications, 196. Birkhäuser Verlag, Basel, 2010. xii+342 pp. ISBN: 978-3-0346-0170-2

Analysis and Mathematical Physics (2021) 11:143 https://doi.org/10.1007/s13324-020-00421-z

The scientific work of Vladimir Maz’ya Alberto Cialdea1 Received: 12 October 2020 / Revised: 22 October 2020 / Accepted: 23 October 2020 / Published online: 27 July 2021 © The Author(s) 2020

1 Introduction Vladimir Maz’ya is an outstanding mathematician whose work had a profound impact on the modern analysis. In more than 60 years of activity, he wrote over 500 papers and 40 monographs. Figure 1 gives an impressive glance of his books. The great depth of the results he has obtained, his fundamental new ideas, and his skilled technique characterize his papers. Among his most important results are the discovery of the equivalence between Sobolev and isoperimetric/isocapacitary inequalities (which we shall describe in Sect. 2), his counterexamples related to Hilbert’s 19th and 20th problems (see Sect. 3), his solution, together with Yuri Burago, of a problem in harmonic potential theory posed by Riesz and Nagy, his solution with Mikhail Shubin of a problem in the

B 1

Alberto Cialdea [email protected] Department of Mathematics, Computer Sciences and Economics, University of Basilicata, V.le dell’Ateneo Lucano, 10, 85100 Potenza, Italy

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s13324-020-00421-z. Chapter 1 was originally published as Cialdea, A. Analysis and Mathematical Physics (2021) 11:143. https://doi.org/10.1007/s13324-020-00421-z. Reprinted from the journal

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Fig. 1 Maz’ya’s books

spectral theory of the Schrödinger operator formulated by Gelfand, his solution with Victor Havin of a problem related to harmonic approximation posed by Mergelyan. He solved also Vladimir Arnol’d’s problem for the oblique derivative boundary value problem and Fritz John’s problem on the oscillations of a fluid in the presence of an immersed body. He gave also seminal contributions to the development of the theory of capacities, nonlinear potential theory, the asymptotic and qualitative theories of arbitrary order elliptic equations, the theory of ill-posed problems, the theory of boundary value problems in domains with piecewise smooth boundary. He introduced the concept of Approximate Approximation, which provides a completely new approach to fast numerical procedures. He received several Honours and Awards: – – – – – – – – – –

1962, Prize of the Leningrad Mathematical Society, Russia. 1990, Doctor honoris causa of the University of Rostock, Germany. 1999, Humboldt Research Prize. 2001, Corresponding Fellow of the Royal Society of Edinburgh. 2002, Member of the Royal Swedish Academy of Sciences. 2003, Verdaguer Prize of the French Academy of Sciences. 2004, The Celsius Gold Medal of the Royal Society of Sciences at Uppsala. 2009, Senior Whitehead Prize of the London Mathematical Society. 2012, Elected Fellow of the American Mathematical Society. 2013, Foreign Member of the Georgian National Academy of Sciences.

His reputation is attested by several Conferences which have been organized in his honour: – Sobolev spaces and potential theory. Conference in honour of Vladimir Maz’ya, Kyoto, Japan, 1993. 2

Reprinted from the journal

The scientific work of Vladimir Maz’ya

– Functional Analysis, Partial Differential Equations and Applications. Conference in honour of Vladimir Maz’ya, Rostock, Germany, 1998. – Analysis, PDEs and Applications. Conference on the occasion of the 70th birthday of Vladimir Maz’ya, Rome, Italy, 2008. – Nordic-Russian Symposium in honour of Vladimir Maz’ya on the occasion of his 70th birthday, Stockholm, Sweden, 2008. – Analysis of Partial Differential Equations, Symposium in honour of Vladimir Maz’ya on the occasion of his 75th Birthday, Liverpool, UK, 2013. – Sobolev Spaces and Partial Differential Equations, on the occasion of the 80th birthday of Vladimir Maz’ya, Accademia dei Lincei, Rome, Italy, 2018. – Harmonic Analysis and PDE, International Conference in honor of Vladimir Maz’ya, Holon, Israel, 2019.

Laurent Schwartz and Vladimir Maz’ya (Paris, 1992)

Several books and papers have been dedicated to him: – Two volumes of “The Maz’ya Anniversary Collection”, edited by Rossmann, J., Takaˇc, P., Wildenhain, Birkhäuser, 1999. – Mathematical Aspects of Boundary Element Methods, dedicated to Vladimir Maz’ya on the occasion of his 60th birthday, edited by M. Bonnet, A.M. Sändig and W. Wendland, Chapman & Hall/CRC Research Notes in Mathematics, London, 1999. – Perspectives in Partial Differential Equations, Harmonic Analysis and Applications: A Volume in Honor of Vladimir G. Maz’ya’s 70th Birthday, edited by D. Mitrea and M. Mitrea, Proc. of Symposia in pure mathematics, Vol. 79, Amer. Math. Soc., Providence (R.I.), 2008. Analysis, Partial Differential Equations and Applications. The Vladimir Maz’ya Anniversary Volume, edited by A. Cialdea, F. Lanzara, P.E. Ricci, Operator Theory, Advances and Applications, Vol. 193, Birkhäuser, Berlin, 2009. – D. Eidus et al, Mathematical work of Vladimir Maz’ya (on the occasion of his 60th birthday), Funct. Differ. Equ. 4 (1997), no. 1–2, pp. 3–11. – M.S.Agranovich et al, Vladimir G. Maz’ya, On the occasion of his 65th birthday, Russian Journal of Mathematical Physics, Vol. 10, No. 3, 2003, pp. 239–244. Reprinted from the journal

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– M. S. Agranovich et al, Vladimir Gilelevich Maz’ya (on his 70th birthday), Russian Math. Surveys 63:1(2008), 189-196. – M.V. Anolik et al, Vladimir Gilelevich Maz’ya (On the Occasion of his 70th Anniversary), Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, no. 4 (2008), 3–6.

Vladimir Maz’ya and John Nash (Beijing, 2002)

One of the characteristic of Maz’ya scientific activity is the great variety of topics. This is only a tentative list of the subjects of his research: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

Equivalence of isoperimetric and integral inequalities Theory of capacities and nonlinear potentials Counterexamples related to the 19th and 20th Hilbert problems Boundary behaviour of solutions to elliptic equations in general domains Non-elliptic singular integral and pseudodifferential operators Degenerating oblique derivative problem Estimates for general differential operators Boundary integral equations Linear theory of surface waves The Cauchy problem for the Laplace equation Theory of multipliers in spaces of differentiable functions Characteristic Cauchy problem for hyperbolic equations Boundary value problems in domains with piecewise smooth boundaries Asymptotic theory of differential and difference equations with operator coefficients Maximum modulus principle for elliptic and parabolic systems, contractivity of semigroups Iterative procedures for solving ill-posed boundary value problems Asymptotic theory of singularly perturbed boundary value problems “Approximate approximations” and their applications 4

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The scientific work of Vladimir Maz’ya

19. 20. 21. 22. 23. 24. 25. 26.

Wiener test for higher-order elliptic equations Spectral theory of the Schrödinger operator Navier-Stokes equations History of Mathematics Mesoscale asymptotic expansion Criteria for the accretivity and form boundedness of second order elliptic equations Eigenfunctions of the Fourier transform Sobolev spaces in unrestricted domains

Clearly, it is impossible to give a complete description of the scientific activity of Vladimir Maz’ya in one paper. I had to make some choices, also considering that certain topics will be discussed in other contributions by other participants of the meeting dedicated to his 80th anniversary. Vladimir Maz’ya has a reputation of being the solver of problems which are generally considered as unsolvable. This is why Fichera once compared him with Santa Rita, the 14th century Italian nun who is the Patron Saint of Impossible Causes. About twenty years ago Vladimir Maz’ya and Sergej Nikol’skij met at ISAAC Conference in Berlin (2001). I remember how Nikol’skij told to Maz’ya that he could not sleep because of a problem he could not solve. Maz’ya asked what problem was. Nikol’skij’s answer was: I will not tell you, otherwise you will solve it !

From left to right: Ennio De Giorgi, Gaetano Fichera, Vladimir Maz’ya and Giorgio Salvini (President of Accademia dei Lincei)

2 Equivalence of isoperimetric/isocapacitary and integral inequalities Sobolev embedding theorems are well known. One of them asserts that if 1 < p < n, the following inequality holds   u L p∗ () ≤ C ∇u L p () + u L p ()

(1)

for any u in the Sobolev space W 1, p (). Here  is a domain in Rn and p ∗ = np/(n − p). This result was proved by Sobolev and later extended to the case p = 1 by Reprinted from the journal

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Gagliardo and Nirenberg (1958). In these papers the domain  satisfies the so-called cone property. This means that each point of the domain is the vertex of a spherical cone with fixed height and angle which is situated inside the domain. The inequality (1) does not hold on an arbitrary domain. For example, it is easy to see that if  = {(x, y) ∈ R2 | 0 < x < 1, 0 < y < x 4 }, the function u(x, y) = 1/x 3 belongs to W 1, 2 (), but not to L 6 (). On the other hand there exist domains showing that cone property is not necessary for the validity of inequality (1). At a very young age, when he was still a fourth year undergraduate student, Vladimir Maz’ya discovered the equivalence between Sobolev embeddings like (1) and isoperimetric and isocapacitary inequalities. The classical isoperimetric inequality states that for any planar domain  with a rectifiable boundary of a fixed lenght L we have 4π A ≤ L 2 ,

(2)

where A is the area of . The equality holds if and only if  is a disk. The n-dimensional generalization of (2) is n−1 (3) (mesn g) n ≤ Cn Hn−1 (∂ g), where g is a domain in Rn with smooth boundary ∂ g and compact closure, and Hn−1 −1/n is the (n − 1)-dimensional area. The constant Cn = n −1 vn is such that (3) becomes equality for any ball (here vn denotes the volume of the unit ball). Maz’ya discovered that Sobolev inequality u L n/(n−1) (Rn ) ≤ Cn ∇u L 1 (Rn ) ,

∀ u ∈ C0∞ (Rn )

holds with the same best constant of the isoperimetric inequality (3). Following this idea, Maz’ya was able to characterize more general inequalities like   u L q (,μ) ≤ C ∇u L p () + u L p () ,

1≤ p≤q

where μ is a measure and  u L q (,μ) =



1/q |u|q dμ .

For example, he proved that the inequality u L q (,μ) ≤ C∇u L 1 () , ∀ u ∈ C0∞ ()

(4)

holds if and only if the following isoperimetric inequality is valid μ(g)1/q ≤ CHn−1 (∂ g), ∀ g  , ∂ g ∈ C ∞

(5)

and the best constants in (4) and (5) coincide. 6

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The scientific work of Vladimir Maz’ya

Let us consider the L p norm of the gradient ( p > 1) instead of the L 1 norm in (4), i.e. (6) u L q (,μ) ≤ D∇u L p () , ∀ u ∈ C0∞ (), q ≥ p ≥ 1. Maz’ya proved that this inequality is equivalent to the “isocapacitary” inequality g  , ∂ g ∈ C ∞ ,

μ(g) p/q ≤ C cap p g,

(7)

where the p-capacity cap p is defined as  cap p F = inf



 |∇ϕ| p d x : ϕ ∈ C0∞ (), ϕ| F ≥ 1 .

More precisely, Maz’ya proved that – if (6) holds, then (7) is true and D ≥ C 1/ p ; – if (7) is valid, then (6) holds and D ≤ p( p − 1)(1− p)/ p C 1/ p . This is only the beginning of the story. The equivalence between integral and isoperimetric/isocapacitary inequalities proved to be extremely fruitful. The Maz’ya book “Sobolev Spaces”, of which a second enlarged edition has been published in 2010, contains deep developments of this idea, which lead to several applications to the solvability of boundary value problems for elliptic equations and to theorems on the structure of the spectrum of the corresponding operators.

3 Counterexamples related to the 19th and 20th Hilbert problem The celebrated theorem proved indipendently by De Giorgi and Nash in 1957 states that every solution u ∈ W 1,2 of the linear elliptic scalar equation ∂i (ai j (x)∂ j u) = 0 with variable real valued bounded coefficients satisfies Hölder’s condition. This result implies the infinite differentiability (or the analyticity) of u. This has turned out to be essential for the complete solution of the 19th Hilbert’s problem. For many years people tried to extend this result to equations of higher order or to systems. It was a kind of shock when Maz’ya proved that the result does not hold for equations of higher order. Let us describe his counterexample which appeared in the paper Maz’ya, V. Examples of nonregular solutions of quasilinear elliptic equations with analytic coefficients. Funkcional. Anal. i Priložen. 2:3, (1968) 53–57 (Russian). English translation: Functional Anal. Appl. 2 (1968), 230–234. Let A be the following fourth order differential operator 

xi x j ux x Au ≡ ν + κ |x|2 i j 2

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 +κ

xi x j u |x|2 7





xi x j

x i x j x k xl +μ u xi x j |x|4

 x k xl

A. Cialdea

where the real constants ν, κ and μ are such that ν > 0, κ 2 < μν. The last condition implies that the operator A is strongly elliptic. Consider the equation Au = 0 in B1 = {x ∈ Rn | |x| < 1}. This can be considered as Euler’s equation for the functional



2

 xi x j xi x j ux x ν(u) + 2κ 2 u xi x j u + μ dx . |x| |x|2 i j 2

B1

The function u(x) = |x|a , where  n a =2− + 2

(n − 1)(κn + μ) n2 − , 4 ν + 2κ + μ

belongs to W 2,2 and is solution of the equation Au = 0 in B1 . If κ = n(n − 2), μ = n 2 , ν = (n − 2)2 + ε (ε > 0) the strong ellipticity condition is satisfied and the exponent a is equal to a(ε) = 2 −

n n + 2 2



ε . 4(n − 1)2 + ε

One can check that, if n > 4 and ε is sufficiently small, the solution |x|a(ε) is not bounded around the origin. This shows that a solution in W 2,2 of the equation Au = 0 does not need to be bounded. In the same paper Maz’ya shows how to construct similar counterexamples for elliptic equations of any order 2l and for quasilinear equations. This is how Maz’ya described the discovery of this counterexample in his book “Differential Equations of my young years” (p.180): Whatever I did to justify analyticity of solutions of variational problems of higher order, nothing worked. I was stealing up to the problem in various ways, but the solution sneaked off. The problem did not want to be solved for the life of me! But one day, feeling desperate I decided to consider concrete examples in order to understand at least something, and almost at once found that the hypothesis of analyticity was wrong – this was not expected by anyone! Nina Nikolaevna Uraltseva was the first whom I showed my counterexamples. She frowned saying “It’s impossible!”, but took my manuscript home and promised to check it. A week later she announced for all to hear at the Big Seminar that I was right. Later Maz’ya with Nazarov and Plamenevskii show that also scalar strongly elliptic second-order differential equations in divergence form with measurable bounded complex coefficients in Rn (n > 4) can have generalized solutions which are not bounded in any neighborhood of an interior point of the domain (see Maz’ya, Nazarov, Plamenevskij: Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains. Vol. I, 2000, pp.391–393). While for n = 2 it is known that a generalized solution has to be Hölder continuous (see Morrey (1938), Trans. Am. Math. Soc.), the cases n = 3 and n = 4 are still unsolved. 8

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This problem is contained in a very interesting collection of open problems which Vladimir Maz’ya recently published (Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations, Integral Equations and Operator Theory, 2018). They concern function theory, functional analysis, theory of linear and nonlinear partial differential equations.

4 Multipliers in Sobolev spaces Vladimir Maz’ya and Tatyana Shaposhnikova developed a deep theory of multipliers in spaces of differentiable functions. By a multiplier acting from one function space S1 into another S2 , they mean a function which defines a bounded linear mapping of S1 into S2 by pointwise multiplication. In their theory, the role of the spaces S1 and S2 is played by Sobolev spaces, Bessel potential spaces, Besov spaces, and the like. In their book Theory of Sobolev Multipliers (the second very much enlarged edition appeared in 2009) they describe this theory by proving a lot of results, like characterization of multipliers, trace inequalities, relations between spaces of Sobolev multipliers and other function spaces, and so on. They provide also several applications to analysis, partial differential and integral equations. As they write in the Introduction of the book, they believe that the calculus of Sobolev multipliers provides an adequate language for future work in the theory of linear and nonlinear differential and pseudodifferential equations under minimal restrictions on the coefficients, domains, and other data. Just to give an example of application of the theory of multiplers, let us consider elliptic boundary value problems in domains with “non-regular” boundaries. Let me introduce some notation. The space of multipliers acting from the Sobolev space W pm into W pl (1 ≤ p < ∞, m ≥ l ≥ 0) is denoted by M(W pm → W pl ). Therefore saying that γ ∈ M(W pm → W pl ) means that the pointwise moltiplication u → γ u defines a linear and continuos operator from W pm into W pl . Let  be an open subset of Rn and let P be the operator

Pu =

(−1)|α| D α (aαβ (x) D β u)

|α|,|β|≤h

where aαβ ∈ C l−h (), l ≥ h. Let us suppose that the Gårding inequality  Re

 |α|=|β|=h

aαβ (x) D α u D β u d x ≥ C u2W h () 2

holds for any u ∈ C0∞ (). This implies that the equation Pu = f with f ∈ W −h () is uniquely solvable in W˚ h (). Reprinted from the journal

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A. Cialdea

The function u is said to be a solution of the generalized Dirichlet problem if ⎧ l ⎪ ⎨u ∈ W p () Pu = f in  ⎪ ⎩ u − g ∈ W pl () ∩ W˚ ph (),

(8)

where f ∈ W pl−2h () and g ∈ W pl () are given. Let us suppose that the boundary of  satisfies a condition expressed in terms of l+1−h−1/ p multipliers. Specifically, let us suppose that ∂ ∈ W p if p(l − h) > n or l+1−h−1/ p ∂ ∈ M p (δ) if p(l − h) ≤ n. The latter means that for each point of the boundary there exists a neighborhood U and a Lipschitz function ϕ such that U ∩  = {(x, y) ∈ U | x ∈ Rn−1 , y > ϕ(x)}

(9)

and ∇ϕ; Rn−1  M W l−1−1/ p ≤ δ . p

Here δ is a small constant and M W ps is the space of multipliers in W ps for s > 0 and L ∞ for s ≤ 0. Under these conditions on the boundary, Maz’ya and Shaposhnikova proved the following existence result: Given f ∈ W pl−2h (), g ∈ W pl (), the BVP (8) has one and only one solution u ∈ W pl (). They show also that a simple sufficient condition which implies their assumption on the boundary when p(l − h) ≤ n is 

1

[ωl−h (t)/t] p dt < ∞ ,

0

where ωl−h is the modulus of continuity of the vector-function ∇l−h ϕ, ϕ being the function in (9).

5 Lp -dissipativity of partial differential operators In a series of joint papers with Vladimir Maz’ya we have considered the problem of characterizing the L p -dissipativity of partial differential operators with complex coefficients. I recall that a linear operator A defined on D(A) ⊂ L p () and with range in L p () is said to be L p -dissipative if  Re



Au, u |u| p−2 d x ≤ 0

for any u ∈ D(A). Here  is a domain in Rn and the functions u are complex valued. 10

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One of our results gives a necessary and sufficient condition for the L p -dissipativity of the partial differential operator with complex valued coefficients Au = div(A ∇u)

(10)

(a hk ∈ L ∞ (),  ⊂ Rn , 1 < p < ∞). Specifically, under the assumption that Im A is symmetric (i.e. Im A t = Im A ), we have proved that A is L p -dissipative if and only if  (11) | p − 2| |Im A (x)ξ, ξ | ≤ 2 p − 1 Re A (x)ξ, ξ  for almost any x ∈ , for any ξ ∈ Rn . Examples show that if Im A is not symmetric or if there are lower order terms, this result is not true. In general condition (11) is only necessary. If Im A is symmetric, (11) is equivalent to the condition 4 Re A (x)ξ, ξ  + Re A (x)η, η − 2(1 − 2/ p)Im A (x)ξ, η ≥ 0 p p for almost any x ∈  and for any ξ, η ∈ Rn . More generally, if the matrix Im A is not symmetric, the condition 4 Re A (x)ξ, ξ +Re A (x)η, η+2( p −1 Im A (x)+ p −1 Im A ∗ (x))ξ, η ≥ 0 p p (12) for almost any x ∈  and for any ξ, η ∈ Rn ( p  = p/( p − 1)) is only sufficient for the L p -dissipativity of A. Recently several authors have considered the class of operators such that the form (12) is not merely non-negative, but strictly positive, i.e. there exists κ > 0 such that 4 Re A (x)ξ, ξ  + Re A (x)η, η + 2( p −1 Im A (x) + p −1 Im A ∗ (x))ξ, η p p ≥ κ(|ξ |2 + |η|2 ) (13) for almost any x ∈  and for any ξ, η ∈ Rn . These operators, which could be called p-strongly elliptic, are playing an increasingly important role in the study of differential operators with complex coefficients, in particular in the study of boundary value problems with L p data. We remark that, if p = 2, condition (13) reduces to the classical strong ellipticity condition ReA (x)ζ, ζ  ≥ κ|ζ |2 for almost any x ∈  and for any ζ ∈ Cn . We have characterized the L p -dissipativity also for other classes of operators. Here I want just to mention a characterization of L p -dissipativity we have obtained for the the system of linear elasticity Eu = u + (1 − 2ν)−1 ∇ div u Reprinted from the journal

11

(14)

A. Cialdea

(ν being the Poisson ratio, ν > 1 or ν < 1/2). In the planar case we proved that the operator E is L p -dissipative if and only if 

1 1 − 2 p

2

2(ν − 1)(2ν − 1) . (3 − 4ν)2

≤

(15)

The condition (15) is necessary for the L p -dissipativity of operator (14) in any dimension, even when the Poisson ratio is not constant. At the present it is not known if condition (15) is also sufficient for the L p -dissipativity of elasticity operator for n > 2, in particular for n = 3. This is another open problem contained in the above mentioned collection “Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations”. Our results on L p -dissipativity can be found in the monograph Cialdea-Maz’ya, Semi-bounded Differential Operators, Contractive Semigroups and Beyond, Operator Theory: Advances and Applications, 243, Birkhäuser, Berlin (2014), where they are considered in the more general frame of semi-bounded operators. Very recently (Cialdea-Maz’ya, Criterion for the functional dissipativity of second order differential operators with complex coefficients, to appear) we have introduced the more general concept of functional dissipativity of the operator (10) with respect to a certain function ϕ. Here ϕ is a positive function defined on R+ such that s ϕ(s) is strictly increasing. Let us denote by  the related Young function  (t) =

t

s ϕ(s) ds .

0

If  Re



A ∇u, ∇(ϕ(|u|) u) d x ≥ 0

for any u ∈ H˚ 1 () such that ϕ(|u|) u ∈ H˚ 1 (), we say that the operator A is functional dissipative or L  -dissipative, in analogy with the terminology used when ϕ(t) = t p−2 . We proved that, if I m A t = I m A , the operator A is L  -dissipative if and only if |s ϕ  (s)| |Im A (x) ξ, ξ | ≤ 2

 ϕ(s) [s ϕ(s)] Re A (x) ξ, ξ 

(16)

for almost every x ∈  and for any s > 0, ξ ∈ R N . As for L p -dissipativity, this condition leads to define a new class of operators, which we shall call -strongly elliptic. They are the second order scalar operators such that [1 − 2 (t)]Re A (x) ξ, ξ  + Re A (x) η, η + [1 + (t)]Im A (x) ξ, η + [1 − (t)]Im A ∗ (x) ξ, η ≥ κ(|ξ |2 + |η|2 ) 12

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for a certain κ > 0 and for almost every x ∈  and for any t > 0, ξ, η ∈ R N . Here  is the function defined by the relation

    s ϕ(s) = −

s ϕ  (s) . s ϕ  (s) + 2 ϕ(s)

6 History of mathematics In 1998 Vladimir Maz’ya and Tatyana Shaposhnikova published the huge volume “Jacques Hadamard. A Universal Mathematician”. It contains more than 500 pages and it was later translated in French and in Russian. It is both a biography and a description of Hadamard’s mathematics. The book is also full of photographs and pictures which make the reading extremely pleasant. The authors made an amazing job finding their sources. As Roger Cooke wrote in “Maz’ya’s work on the biography of Hadamard” (in Rossmann et al. (eds.), The Maz’ya anniversary collection (Birkhäuser Verlag, Basel, 1999)) although much of the relevant material unfortunately disappeared during World War II, they were able to find a great deal of new material nevertheless. They located Hadamard’s grandson, Francis Picard and were able to look at the materials collected by Hadamard’s daughter Jacqueline, including her autobiography. They interviewed several mathematicians who had known Hadamard personally, and they submitted their drafts of the new and revised material to experts in a number of areas of mathematics and the history of mathematics, and the history of French mathematics in particular. The result, published in English this time, was a triumph of mathematical biography that one can only hope will be repeated for many other great twentieth-century mathematicians. …It will probably turn out to be the definitive biography of Hadamard. Another book written by Maz’ya and recently translated in russian, which gives an important contribution to the history of mathematics is “Differential Equations of My Young Years”. It is an autobiography which covers events from 1937 till 1968. The book is full of wonderful photographs. Reading this book means taking a dip in the past. It gives a precise description of the Soviet life in 40s-60s of the last century. Even if the book is full of mathematics, it can be read by anyone. As Havin writes in the foreword, the fact that mathematics appears on many of its pages in no way diminishes the book’s clarity of discourse and attraction to a variety of readers. In addition to more personal events from adolescence and early adulthood, we can read very interesting facts about the russian mathematics in that period.

7 Other books Although Maz’ya’s books appearing in Fig. 2 do not concern his scientific activity, I think that they deserve a mention, because they are another proof of the great imagination he has. Reprinted from the journal

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A. Cialdea

Fig. 2 Maz’ya’s fairy tales

They are three books of fairy tales intended for middle school children. The heroes of these stories are animals and people, as well as space aliens and underground inhabitants. As Gohberg wrote in “Vladimir Maz’ya: friend and mathematician. Recollections.” (in Rossmann et al. (eds.), The Maz’ya anniversary collection (Birkhäuser Verlag, Basel, 1999)): whatever he writes is beautiful, his love for art, music and literature seeming to feed his mathematical aesthetic feeling. Funding Open access funding provided by Università degli Studi della Basilicata within the CRUI-CARE Agreement.

Compliance with ethical standards Conflict of interest The author declares that there is no conflict of interest. Availability of data and materials Not applicable. The manuscript has no associated data. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Analysis and Mathematical Physics (2022) 12:54 https://doi.org/10.1007/s13324-022-00666-w

Topological mappings of finite area distortion Elena Afanas’eva1 · Anatoly Golberg2 Received: 17 January 2022 / Accepted: 15 February 2022 / Published online: 18 March 2022 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2022

Abstract We study the interplay of mappings of finite area distortion (FAD) with finitely biLipschitz mappings, ring and lower Q-homeomorphisms, and absolutely continuous n, p homeomorphisms of the class AC on Riemannian manifolds. Some additional relations to the hyper Q-homeomorphisms, η-quasisymmetric and ω-quasimöbius mappings are also established. As applications of the above results, we provide several extension conditions to the weakly flat and strongly accessible boundaries under FAD-homeomorphisms. Keywords Riemannian manifolds · Mappings of finite area distortion · Finitely bi-Lipschitz homeomorphisms · Quasisymmetry · Quasiconformality · Quasimöbius mappings · Q-homeomorphisms · Moduli of families of curves and surfaces · Boundary behavior of FAD-homeomorphisms · Sobolev classes · Absolute continuity Mathematics Subject Classification Primary: 30L10 · 26B30; Secondary: 30C65 · 53B20

1 Introduction In the paper, we continue the studying main relationships between various classes of mappings whose definitions rely on metric approaches and techniques: finitely bi-Lipschitz mappings, quasisymmetric mappings, quasimöbius and quasiconformal Dedicated to Vladimir Maz’ya, distinguished mathematician and prominent person.

B

Anatoly Golberg [email protected] Elena Afanas’eva [email protected]

1

Institute of Applied Mathematics and Mechanics of the NAS of Ukraine, 1 Dobrovol’skogo St., Slavyansk 84100, Ukraine

2

Department of Mathematics, Holon Institute of Technology, 52 Golomb St., P.O.B. 305, 5810201 Holon, Israel

Chapter 2 was originally published as Afanas’eva, E. & Golberg, A. Analysis and Mathematical Physics (2022) 12:54. https://doi.org/10.1007/s13324-022-00666-w. Reprinted from the journal

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E. Afanas’eva, A. Golberg

mappings, mappings of finite metric distortion and mappings of finite area distortion. The latter is the central object in the present manuscript. We also involve classes of mappings which are called the ring, lower and hyper Q-homeomorphisms and are defined purely geometrically. The interplay between the above classes of mappings allows us to investigate some boundary correspondence problems related to the weakly flat and strongly accessible boundaries. The analytic properties of homeomorphisms of finite area distortion (FAD-homeomorphisms) and their counterparts are of special interest. We show that such n, p homeomorphisms belong to the class AC of absolutely continuous mappings by Malý [23], Hencl [18] and Bongiorno [6]. By its properties, this class is closely related to the standard Sobolev class W 1, p . For detailed information on Sobolev classes, we refer to monograph [26]. Since our studying relates to Riemannian manifolds, the corresponding main definitions and notations can be found in our previous paper [3], although some of them are also given here. The paper’s structure is following. In Sect. 2 we provide the definitions of the above classes of mappings that rely on metric approach. Some main relations between the mentioned classes are also presented. We discuss the modular approach and the corresponding classes of mappings in Sect. 3. The main results related to FAD-homeomorphisms are proven in Sect. 4. They describe the connections between FADhomeomorphisms, absolute continuity, Sobolev classes and geometric representatives given in the previous section. The main analytic features of quasimöbius mappings and their relation to FAD-homeomorphisms are given in Sect. 5. Several applications to the boundary correspondence of FAD-homeomorphisms are provided in Sect. 6. For other results and some illustrating examples we refer again to [3].

2 Terminology and auxiliary results related to metric approach Here we provide the needed definitions related to the metric approach and emphasize the connections between these notions. 2.1 Riemannian manifolds We start with some notions connected with Riemannian geometry. For other definitions and descriptions we refer to our recent paper [3]. Here and throughout the paper we use the following terminology. We denote by M a Riemannian manifold, i.e. an n-dimensional differentiable connected manifold (C ∞ , n ≥ 2), equipped with a Riemannian metric. Let exp p (exponential map) be a diffeomorphism of a neighborhood V of the origin in T p M and exp p (V ) = U , where U is a neighborhood of the point p ∈ M.

16

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The length of a piecewise smooth curve γ : [a, b] → M in local coordinates is determined by

(γ ) :=

b 

gi j (x(γ (t)))x˙ i (t)x˙ j (t)dt,

a d where x˙ i (t) := dt (x i (γ (t))). Recall that gi j (x) is a positive definite symmetric tensor field defined on the local coordinates and obeying the transition rule k l gi j (x) = h kl (y) ∂∂ yx i ∂∂xy j , where k, l = 1, . . . , n, are the so-called dummy indices over which the summation is performed. The n-dimensional volume element is determined by

dvg =

 detgi j d x 1 . . . d x n ,

and it is invariant on M similarly to the length element. Recall that for an arbitrary point p of a Riemannian manifold M, there are its neighborhoods U and the corresponding local coordinates in U such that every geodesic sphere centered at p can be associated with the Euclidean sphere of the same radii centered at the origin, and a bundle of geodesic curves originating from p is associated with a bundle of beams originating from the origin in Rn . As usual we will call these neighborhoods and coordinates normal, see, e.g. [22, Lemma 5.10 and Corollary 6.11]. In normal coordinates, any geodesic sphere has a natural smooth parametrization by the direction cosines of the corresponding rays going from the origin. Moreover, the metric tensor in these coordinates coincides with the identity matrix at the origin; see, e.g., Proposition 5.11 in [22]. Throughout the paper, let D and D  be two domains in connected smooth Riemannian manifolds (M, g) and (M , g  ) of geodesic distances d and d  , respectively. If Br (0) lies with its closure in a neighborhood V of the origin in Rn , we call exp p Br (0) = Br ( p) = B( p, r ) to be a normal ball (or geodesic ball) centered at p of the radius r . We also assume that the closed ball B( p, r ), spheres S( p, ri ) and ring A = A( p, r1 , r2 ), 0 < r1 < r2 < r , centered at p lie in a normal neighborhood U ( p), and, therefore, A( p, r1 , r2 ) = {x ∈ M: r1 < d(x, p) < r2 } and S( p, ri ) = {x ∈ M: d(x, p) = ri }, (i = 1, 2) become the geodesic ring and geodesic spheres, respectively. One of fruitful tools for studying Riemannian manifolds relies on a partition of unity. Recall that the partition of unity, subordinated to the cover {Ui }i∈I of a manifold M, is a collection ϕi : M → R of C ∞ functions (where I is an arbitrary index set, not assumed countable) such that (i) 0 ≤ ϕi (x) ≤ 1 for any point x of M; (ii) the collection of supports {suppϕi }i∈I is locally finite, i.e. for any x ∈ M there is aneighborhood which intersects with a finite number of sets of this collection; (iii) i ϕi ≡ 1 for any point x of M; (iv) suppϕi ⊂ Ui for all i.

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2.2 Mappings of finite metric distortion The Lipschitz mappings are classical representatives of the metric approach in function theory. Recall that a mapping f : M → M is Lipschitz if there exists a constant L < ∞ such that d  ( f (x), f (y)) ≤ L d(x, y), for all x, y ∈ M. We say that a mapping f : M → M is bi-Lipschitz (or L-bi-Lipschitz) if it satisfies the condition 1 d(x, y) ≤ d  ( f (x), f (y)) ≤ Ld(x, y), L for some L ≥ 1 and all x and y in M. Obviously, for L = 1, the mapping f becomes isometric. Following [25], we say that f : D → D  is finitely Lipschitz if L(x, f ) < ∞ for all x ∈ D and finitely bi-Lipschitz if 0 < l(x, f ) ≤ L(x, f ) < ∞ for all x ∈ D, where L(x, f ) := lim sup

d  ( f (x), f (y)) , d(x, y)

(1)

l(x, f ) := lim inf

d  ( f (x), f (y)) . d(x, y)

(2)

y→x

and

y→x

Note that the Euclidean and Riemannian (geodesic) distances are equivalent in local coordinates (cf. [22, Lemma 6.2]). Obviously any Lipschitz (bi-Lipschitz) mapping is finitely Lipschitz (finitely biLipschitz) but not vice versa. We say that a mapping f : D → D  is of finite metric distortion, abbr. f ∈ FMD, if f admits the Lusin (N )-property with respect to the n-dimensional Hausdorff measure and 0 < l(x, f ) ≤ L(x, f ) < ∞ almost everywhere (a.e.), cf. [25]. 2.3 FMD and finitely bi-Lipschitz mappings The notion of hyperconvexity relates to a wider class of spaces and becomes a crucial tool for establishing our results, since it allows us to apply a Kirszbraun type theorem on extending Lipschitz mappings. Recall that a metric space M is called hyperconvex if ∩α∈ B(xα , rα ) = ∅ for any collection of points {xα }α∈ in M and of positive numbers {rα }α∈ such that d(xα , xβ ) ≤ rα + rβ for any α and β in ; see [12]. The following auxiliary result is inverse to [3, Lemma 5.1]. It holds since in contrast to the referred statement here the condition on l(x, f ) and L(x, f ) is strengthened. Proposition 2.1 Suppose that M is a hyperconvex space. Then any F M D-homeomorphism f : D → D  , satisfying the inequality 0 < l(x, f ) ≤ L(x, f ) < ∞ for all x ∈ D, is finitely bi-Lipschitz. Note that finitely Lipschitz mappings preserve the sets of zero Hausdorff measure on Riemannian manifolds in both directions, or equivalently, the Lusin (N ) and (N −1 )18

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properties, see Theorem 3.1 and Corollary 3.1 in [3]. Thus, the above proposition directly follows from definitions. Combining Lemma 5.1 from [3] with Proposition 2.1 yields Proposition 2.2 Suppose that M is a hyperconvex space. Then the following statements hold: (i) every finitely bi-Lipschitz homeomorphism f : D → D  is an F M D-homeomorphism f : D → D  . (ii) any F M D-homeomorphism f : D → D  , satisfying 0 < l(x, f ) ≤ L(x, f ) < ∞ for all x ∈ D, is a finitely bi-Lipschitz homeomorphism. 2.4 Mappings of finite area distortion Although any FMD-mapping possesses the Lusin (N )-property with respect to the n-dimensional Hausdorff measure, for mappings of finite area distortion we add two requirements related to preserving k-dimensional areas; see, [21] (cf. [25]). We say that a mapping f : D → D  has the (Ak )-property if the following two conditions hold: (1) (Ak ): for a.e. k-dimensional surface S in D, the restriction f | S has the (N )-property with respect to area; (2) (Ak ): for a.e. k-dimensional surface S  in D  = f (D), the restriction f | S has the (N −1 )-property for each lifting S of S  with respect to area. Here a surface S in D is a lifting of a surface S  in D  under a mapping f : D → D  if S  = f ◦ S. Recall that a mapping f : D → D  is of finite area distortion in dimension k = 1, . . . , n − 1, abbr. f ∈ FADk , if f ∈ FMD and has the (Ak )-property. We also say that a mapping f : D →  is of finite area distortion, abbr. f ∈ FAD, if f ∈ FADk for every k = 1, . . . , n − 1. The following result provides the connection between finitely bi-Lipschitz and FAD-homeomorphisms. Theorem 2.1 Let M be a hyperconvex space. Then the following statements hold: (i) if f : D → D  is a finitely bi-Lipschitz homeomorphism possessing the (A(1) k ) and )-properties, then f is an FAD -homeomorphism for some k = 1, . . . , n − 1; (A(2) k k (ii) if f : D → D  is an FAD-homeomorphism satisfying the inequality 0 < l(x, f ) ≤ L(x, f ) < ∞ for all x ∈ D, then f is a finitely bi-Lipschitz homeomorphism. Proof (i) Since M is a hyperconvex space, due to [3, Theorem 3.1 and Remark 3.1], any finitely Lipschitz homeomorphism f : D → D  preserves the sets of zero Hausdorff measure in smooth connected Riemannian manifolds, or equivalently, the Lusin (N ) and (N −1 )-properties. Applying [3, Lemma 5.1] to f yields that every finitely bi-Lipschitz homeomorphism f : D → D  possessing the Lusin (N )-property with respect to the k-dimensional Hausdorff measure is an FMD-homeomorphism f : D → D  , k = 1, . . . , n. And, therefore, f : D → D  is an FADk -homeomorphism for some k = 1, . . . , n − 1. Reprinted from the journal

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The second part (ii) directly follows from Proposition 2.1 and definitions of FAD 

and FADk -homeomorphisms. 2.5 Quasisymmetric mappings Another extension of Lipschitzness which is closely related to the class of quasiconformal mappings and employs a metric approach provides the class of quasisymmetric homeomorphisms. Let η: [0, ∞) → [0, ∞) be a homeomorphism. A homeomorphism f : D → D  is called η-quasisymmetric (abbr. η-QS homeomorphism) if the inequality   d  ( f (x), f (y)) d(x, y) ≤ η d  ( f (x), f (z)) d(x, z) holds for any triple x, y, z ∈ D, x = z, (see, e.g. [39]). Recall that any bi-Lipschitz homeomorphism is also an η-QS homeomorphism; see, e.g. [38]. A weaker condition than η-QS is called H -quasisymmetry. More precisely, a mapping f is a weakly H -quasisymmetric homeomorphism (abbr. weakly H -QS homeomorphism) if there is a constant H ≥ 1 such that d(x, y) d  ( f (x), f (y)) ≤H  d ( f (x), f (z)) d(x, z) for any triple x, y, z ∈ D, x = z; cf. [16]. In general, weakly quasisymmetric mappings are not η-quasisymmetric; see [16, p. 79]. 2.6 Quasimöbius mappings n

The class of quasimöbius mappings has been naturally arisen in R = Rn ∪ {∞} as an extension of quasisymmetry by removing a preservation of the point of infinity; see [37]. Its definition employs also the metric approach. Let ω(t) be a distortion function. A topological embedding f : X → Y of a metric space is called ω-quasimöbius (ω-QM) if for any four distinct points x1 , x2 , x3 , x4 ∈ X , the following inequality   d  ( f (x1 ), f (x2 )) · d  ( f (x3 ), f (x4 )) d(x1 , x2 ) · d(x3 , x4 ) ≤ ω d  ( f (x1 ), f (x3 )) · d  ( f (x2 ), f (x4 )) d(x1 , x3 ) · d(x2 , x4 ) holds. 2.7 Metrically quasiconformal mappings Going back to [17] we recall the metric definition of quasiconformality. 20

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Given a homeomorphism f from a metric space X to a metric space Y , then for x ∈ X and r > 0 set H f (x, r ) =

sup{d  ( f (x), f (y)): d(x, y) ≤ r } . inf{d  ( f (x), f (y)): d(x, y) ≥ r }

(3)

A homeomorphism f : X → Y is called metrically K -quasiconformal if there is a constant K < ∞ so that lim sup H f (x, r ) ≤ K for all x ∈ X . r →0

3 Modulus of surface families and related classes of mappings In this section, we recall the notion of modulus of surface families and the classes of mappings called the ring Q-homeomorphisms, lower Q-homeomorphisms and hyper Q-homeomorphisms. In contract to the previous section all such mappings have purely geometric (modular) description, namely, for such mappings the modulus of the corresponding surface/curve families is controlled from below/above by integrals depending on a measurable function Q and arbitrary admissible metrics. 3.1 Modulus of surface families For a Borel function ρ : M → [0, ∞], its integral over the surface S is determined by 

 ρ dAg :=

ρ(y) N (S, y)d H k y, M

S

where N (S, y) stands for the multiplicity function of S, measurable with respect to the arbitrary Hausdorff measure H k ; see, e.g. [33, Theorem II (7.6)]. A Borel function ρ: M → [0, ∞] is called admissible for the family of kdimensional surfaces in M, k = 1, 2, ..., n − 1, abbr. ρ ∈ adm , if  ρ k dAg ≥ 1 ∀ S ∈ .

(4)

S

Here and later on, by the k-dimensional Hausdorff area of a Borel set B in M (or simply area of B in the case k = n − 1) associated with the surface S: ω → M, we mean  A S (B) = AkS (B) := N (S, y) d H k y, B

(cf. [13, Ch. 3.2.1]). The surface S is called rectifiable (quadrable), if A S (M) < ∞ (see, e.g [25, Ch. 9.2]). Reprinted from the journal

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The modulus of family is defined by  M( ) :=

ρ n (x) dvg .

inf

ρ∈adm D

We say that a property P holds for almost all (a.a.) S ∈ , if the modulus of a subfamily ∗ of S ∈ , for which P fails, equals zero. Following [25], a Borel function ρ: M → [0, ∞] is called extensively admissible for the family of k-dimensional surfaces S in M, abbr. ρ ∈ ext adm , if the admissibility condition (4) is fulfilled only for a.a. S ∈ . 3.2 Ring and lower Q-homeomorphisms Let Q: M → (0, ∞) be a measurable function. A homeomorphism f : D → D  is called a lower Q-homeomorphism at a point x0 ∈ D, if there exists δ0 ∈ (0, d(x0 )), d(x0 ) := supx∈D d(x, x0 ), such that for any ε0 < δ0 and every geodesic ring Aε = A(x0 , ε, ε0 ) = {x ∈ M: ε < d(x, x0 ) < ε0 }, ε ∈ (0, ε0 ), the lower bound  M( f (ε )) ≥

inf

ρ∈ext adm ε

ρ n (x) dvg Q(x)

D∩Aε

holds. Here and hereafter, we denote by ε the family of all intersections of geodesic spheres S(x0 , r ) = {x ∈ M: d(x, x0 ) = r }, r ∈ (ε, ε0 ), centered at x0 ∈ D with the domain D. We also say that a homeomorphism f : D → D  is a lower Q-homeomorphism, if f is a lower Q-homeomorphism at each point x0 ∈ D (Fig. 1). Later on, for sets A, B and C from M, by (A, B; C) we denote a set of all curves γ : [a, b] → M, which join A and B in C, i.e. γ (a) ∈ A, γ (b) ∈ B and γ (t) ∈ C for all t ∈ (a, b).

Fig. 1 Lower homeomorphism

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Let Q: M → (0, ∞) be a measurable function. We say that a homeomorphism f : D → D  is a ring Q-homeomorphism at a point x0 ∈ D if the following bound  M ( f (C), f (C0 ); D  ) ≤

 Q(x) · ηn (d(x, x0 )) dvg

(5)

A∩D

is fulfilled for any geodesic ring A = A(x0 , ε, ε0 ), 0 < ε < ε0 < ∞, any two continua (compact connected sets) C ⊂ B(x0 , ε) ∩ D and C0 ⊂ D\B(x0 , ε0 ) and any Borel function η: (ε, ε0 ) → [0, ∞] such that ε0 η(r ) dr ≥ 1.

(6)

ε

We also say that f is a ring Q-homeomorphism in D if (5) is true for all points x0 ∈ D (Fig. 2). For an extension of ring homeomorphisms and the corresponding results, we refer to [9]. The following statement formulated for the case of Finsler manifolds can be found in [2, Theorem 3.1]. Note that Riemannian manifolds provide a partial case of Finsler ones. Proposition 3.1 Let D and D  be two domains in M and M , and Q: D → (0, ∞) be integrable with the degree n − 1 in some normal neighborhood of a point x0 ∈ D. Suppose that f : D → D  is a lower Q-homeomorphism at x0 , then f is a ring Q ∗ homeomorphism at x0 with Q ∗ (x) = c · Q n−1 (x), where the constant c is arbitrarily close to 1.

Fig. 2 Ring homeomorphism

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3.3 Hyper Q-homeomorphisms A simultaneous use of similar upper integral bounds for the moduli of surfaces in the image and in the inverse image provides a class of hyper Q-mappings. Let  be an open set in D. Given a pair Q(x, y) = (Q 1 (x), Q 2 (y)) of measurable functions Q 1 :  → [1, ∞] and Q 2 :  → [1, ∞], we say that a mapping f :  →  , f () =  , is a hyper Q-mapping in dimension k = 1, ..., n − 1, if  M( f ( )) ≤

Q 1 (x) · n (x) dvg 

and  M( ) ≤

Q 2 (y) ·  (y) dvg n



for every family of k-dimensional surfaces S in  and all  ∈ adm and  ∈ adm f ( ). We also say that a mapping f :  →  is a hyper Q-mapping if f is a hyper Q-mapping in all dimensions k = 1, ..., n − 1; cf. [25]. 3.4 Capacities of condensers Another crucial tool in our research relates to the notion of capacities of condensers. Following [17], recall that a Borel function ρ: U → [0, ∞] is a weak gradient of u in an open set U if  ρ ds,

|u(x) − u(y)| ≤ γ

whenever γ is a rectifiable curve joining two points x and y in U . Suppose that E and F are closed subsets of U in M. The triple (E, F; U ) is called a condenser, and its conformal capacity is defined by  cap(E, F; U ) = inf

ρ n dvg , U

where the infimum is taken over all weak gradients of all functions u in U such that u| E ≥ 1 and u| F ≤ 0. Such a function u is called admissible for the condenser (E, F; U ). The capacity of condenser (E, F; U ) is equal to the modulus of (E, F; U ), cap(E, F; U ) = M((E, F; U )), 24

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Topological mappings of finite area distortion

where (E, F; U ) denotes the family of all curves in U joining two disjoint closed subsets E and F of U ; see, e.g. [17, Proposition 2.17] for metric spaces. Some fruitful estimates for the capacity can be found in [26,27]. We also need the following lower bound [24] for a ring condenser E = (G, E) ⊂ D, (capE)n−1 ≥ c

diamn E , vg G

(8)

where G ⊂ D is open, E is a connected compactum contained in G, and c is a positive constant depending only on n. Note that this estimate is proven for Rn . However, we apply this bound in normal neighborhoods of a point of Riemannian manifold, where the Euclidean and Riemannian structures agree. In terms of the condenser (E, F; U ), E = (E, D\G; G). 3.5 Finite Lipschitzness of ring Q-homeomorphisms Here we provide some relations between the homeomorphisms defined by the above modular conditions and mappings determined metrically in the previous section. Theorem 3.1 Let D and D  be two domains in Riemannian manifolds M and M , respectively. Suppose that f : D → D  is a ring Q-homeomorphism with Q ∈ L 1loc (D). Then L(x, f ) < ∞ a.e. in D. Proof Let x0 ∈ D be an arbitrary point and A = A(x0 , r , 2r ) ⊂ D be a geodesic ring with sufficiently small r > 0. Since f is a ring Q-homeomorphism,    Q(x) · ηn (d(x, x0 )) dvg , M ( f (C), f (C0 ); D ) ≤ A

where η satisfies the condition (6). (7) between the capacity of ring condenser f (E) = 

Taking into account the relation f (B(x0 , 2r )), f (B(x0 , r )) and the modulus from the last inequality together with the lower bound (8), one obtains

diamn f (B(x0 , r )) c vg f (B(x0 , 2r ))

1/(n−1)

 ≤

Q(x) · ηn (d(x, x0 )) dvg . A

Letting η(d(x, x0 )) = 1/r in A ∩ D and η(d(x, x0 )) = 0 otherwise that satisfies (6), we rewrite the last inequality as ⎛  1/n ⎜ diam f (B(x0 , r )) ≤ c1 r 1−n vg f (B(x0 , 2r )) ⎝



B(x0 ,2r )

where c1 is a positive constant depending only on n. Reprinted from the journal

25

⎞(n−1)/n ⎟ Q(x) dvg ⎠

,

E. Afanas’eva, A. Golberg

A given open set G consider two set functions (G) = vg f (G). Then the following estimate holds diam f (B(x0 , r )) ≤ c2 r



(B(x0 , 2r )) vg B(x0 , 2r )

1/n 

 G

Q(x) dvg and (G) =

(B(x0 , 2r )) vg B(x0 , 2r )

(n−1)/n

.

Since due to [30] both the limits lim

r →0

(B(x0 , 2r )) =:   (x0 ) and vg B(x0 , 2r )

lim

r →0

(B(x0 , 2r )) =:  (x0 ) vg B(x0 , 2r )

exist and are finite a.e., we obtain  1/n   (n−1)/n L(x0 , f ) ≤ c2   (x0 )  (x0 ) , where c2 > 0 is a constant depending on n only. This completes the proof.



By Stepanoff’s theorem [34] we have Corollary 3.1 Any ring Q-homeomorphism f : D → D  is differentiable a.e. in D. Applying the above theorem to both the homeomorphism f and its inverse we derive Theorem 3.2 Let f be a hyper ring Q-homeomorphism with Q 1 ∈ L 1loc (D) and Q 2 ∈ L 1loc (D  ). Then 0 < l(x, f ) ≤ L(x, f ) < ∞ a.e. in D. Moreover, f and f −1 possess the Lusin (N ) and (N −1 ) properties with respect to any k-dimensional Hausdorff measure, k = 1, . . . , n. Proof Note that l(x, f ) > 0 implies the Lusin (N −1 )-condition, whereas L(x, f ) < ∞ provides l( f (x), f −1 ) > 0, and, therefore, the Lusin (N )-property with respect to the n-dimensional Hausdorff measure; cf. [29]. The extension to all k immediately follows from the lower estimates for Jacobians of order k. Indeed, the Jacobian of order k of f at x is estimated from below by l k (x, f ) a.e. The same is true for the inverse mappings and their Jacobians. 

Corollary 3.2 If f is a hyper ring Q-homeomorphism with Q 1 ∈ L 1loc (D) and Q 2 ∈ L 1loc (D  ), then it is an FAD-homeomorphism.

4 Main results for FAD-homeomorphisms In this section, we formulate the main results on FAD-homeomorphisms between Riemannian manifolds starting from the analytic features. 26

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4.1 Absolute continuity 1, p

Recall that a continuous mapping f in Rn belongs to the class Wloc , iff f ∈ ACL p , i.e. the mapping f is locally absolutely continuous on a.a. segments parallel to the coordinate axes, and the first partial derivatives of f are locally integrable in a degree p; see, e.g. [19, Theorem A.15]. We define an analogue of ACL p , p ≥ 1, for mappings f : M → M which belong to ACn (determined below) and whose partial derivatives are locally integrable with the degree p. The class ACn of absolutely continuous functions of several variables has been introduced by Malý [23] in Rn . Let  ⊂ Rn be an open set. A function f :  → R is n-absolutely continuous (shortly ACn ) if for each ε > 0 there exists δ > 0 such that for every pairwise disjoint finite collection {B(x j , r j )} of balls in , we have 

 (osc B(x j ,r j ) f )n < ε,

m B(x j , r j ) < δ ⇒

j

(9)

j

where osc B f = sup{| f (x) − f (y)|: x, y ∈ B} for a ball B. Clearly, any nabsolutely continuous mapping is continuous, and n-absolute continuity of a mapping f = ( f 1 , . . . , f n ) is equivalent to n-continuity of each real valued function f j , j = 1, . . . , n. The balls in Malý’s definition cannot be replaced by cubes. To avoid this dependence on shapes, Hencl [18] suggested instead of the implication (9) the following condition 

m B(x j , r j ) < δ ⇒

 (osc B(x j ,λr j ) f )n < ε,

j

(10)

j

for some fixed λ, 0 < λ ≤ 1. This class is denoted by ACnλ . Note that the implication (10) provides a weaker condition than (9) for both cases of balls and cubes. Moreover, the mappings satisfying (10) form a regular subclass of 1,n , and it is stable under quasiconformal mappings. Wloc The following fruitful extension of Hencl’s definition having essentially deep properties has been introduced in [6]. By this approach, the constant λ is replaced by a function λ:  → (0, 1]. Denote by  a family of all such functions λ. We say that a mapping f :  → Rm is n, -absolutely continuous (shortly ACn ) if for each ε > 0 there exists δ > 0 such that for every pairwise disjoint finite collection {B(x j , r j )} of balls in , we have 

m B(x j , r j ) < δ ⇒

j

 (osc B(x j ,λ(x j )r j ) f )n < ε.

(11)

j

The following important properties characterize the class ACn . First, ACnλ is a 1,n 1,n proper class of ACn intersected with Wloc . Secondly, ACn \Wloc = ∅. Note also that ACn is stable under a large class of mappings including the quasiconformal ones; see [6]. Reprinted from the journal

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E. Afanas’eva, A. Golberg n, p

We denote a class AC of mappings which belong to ACn and whose partial derivatives are integrable with the degree p, p ≥ 1. Clearly, that the condition (11) can be easily extended to mappings f : X → Y between two metric measure spaces (X , d, μ) and (Y , d  , ν) by the following way. A μ-measurable mapping f :  → Y is n, -absolutely continuous if for every ε > 0 there existsδ > 0 such that for each finite collection of nonoverlapping balls B j satisfying j μ(B j ) < δ, we have  n diam B(x j ,λ(x j )r j ) f < ε,

(12)

j

where  ⊂ X is an open set and  is a collection of functions λ:  → (0, 1]. Theorem 4.1 Let D and D  be two domains in connected smooth Riemannian manifolds M and M , respectively. Suppose that f : D → D  is a finitely Lipschitz homeomorphism with L(x, f ) ∈ L 1loc (D), then f ∈ ACn,1  (D). Proof By [7, Theorem 3.1, Ch. V] we pass to a metric space (X , d, μ), since M is a connected Riemannian manifold. Due to (1), for any ε > 0 there is δ f > 0 such that for any y ∈ B(x, δ) we have d  ( f (x), f (y))/d(x, y) < L(x, f ) + ε. Pick ε = 1, one gets d  ( f (x), f (y)) < (L(x, f ) + 1) d(x, y).

(13)

 Consider a finite collection of nonintersecting balls B(x j , r j ) in D such that j μ(B j ) < δ. Then applying the triangle inequality, n-regularity by Ahlfors of Riemannian manifolds and (13), we have  n  diam B(x j ,λ(x j )r j ) f = j

n

sup

B(x j ,λ(x j )r j )

j

d  ( f (x), f (y))

 n 2(L(x j , f ) + 1)λ(x j )r j < j

≤2 C n



μ(B j ) < ε,

j

where λ(x j ) = 1/(L(x j , f ) + 1) ∈ (0, 1], C ≥ 1 is a constant from the n-regularity condition and δ = min{ε/(2n C), δ j }. Here δ j is δ f for which (13) holds at x j . Thus, 

f ∈ ACn . Assuming the local integrability of L(x, f ) completes the proof. Remark 4.1 The exponent 1 in Theorem 4.1 is sharp (w.r.t. ACn,1  ) and cannot be replaced by any larger real number. Consider a homeomorphism   x 1−c , 0 < c < 1, f (x) = x1 , . . . , xn−1 , n 1−c 28

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between the unit open cube (0, 1)n and n-dimensional interval (0, 1)n−1 ×(0, 1/(1−c)) of Rn . By a direct calculation, l(x, f ) = 1 and L(x, f ) = xn−c , and, therefore, f is finitely Lipschitz with locally integrable L(x, f ) for any c, 0 < c < 1. However, 1

 L (x, f ) dm(x) = p

(0,1)n

−cp

xn

d xn = ∞

0

for any p ≥ 1/c. Thus, one can formulate the following result. Theorem 4.2 There exists a finitely Lipschitz homeomorphism f of locally integrable n,1+ε for any positive ε. L(x, f ) such that f fails to belong to AC Remark 4.2 Moreover, there exist finitely Lipschitz homeomorphisms which do not belong to ACn,1  . Here we extend the example given in [20] for the planar case to higher dimensions. Consider an automorphism of the unit ball Bn in Rn of such a form f (x) =

1 1 1 − x2 sin 2 , x1 sin 2 2 2 + x2 x1 + x2 x1 + x22

1 +x2 cos 2 , x3 , . . . , xn x1 + x22 x1 cos

x12

(15)

for x = 0 and defined at the origin by a continuity, i.e. f (0) = 0. This mapping provides a rotation in the plane x1 , x2 with the angle θ = 1/(x12 + x22 ) which tends to infinity as both x1 , x2 approach 0. By a straightforward calculation, l(x, f ) =

 1 + θ 2 − θ,

L(x, f ) =



1 + θ 2 + θ,

whereas the rest of stretchings are equal to 1. Hence, J (x, f ) = 1, and f is a diffeomorphism in the punctured ball. Moreover, f satisfies the Lusin (N ) and (N −1 )-properties with respect to each Hausdorff measure in Bn . So, f is an FAD-homeomorphism. However, the partial derivatives ∂ f i /∂ x j , i, j = 1, 2, fail to be locally integrable in any neighborhood of points x = (0, 0, x3 , . . . , xn ) ∈ Bn . Remark 4.3 On the other hand, a finitely Lipschitz homeomorphism (with unbounded n, p L(x, f ) from above) can belong to AC for some p > 1. Indeed, consider a quasiconformal automorphism of the unit ball Bn in Rn of such a form, f (x) = x|x|α−1 , 0 < α < 1,

f (0) = 0.

As any quasiconformal mapping it belongs to W 1, p , where p > n and depends only on the coefficient of quasiconformality and n. Here the coefficient of quasiconformality Reprinted from the journal

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E. Afanas’eva, A. Golberg

is equal to 1/α, L(x, f ) = |x|α−1 , l(x, f ) = α|x|α−1 . Thus, f is finite Lipschitz in Bn , and its inverse is 1/α-Lipschitz in the same ball. Note that 

1 L (x, f ) dm(x) = ωn−1 p

Bn

r (α−1) p+n−1 dr < ∞,

0

if p < n/(1 − α). Here ωn−1 denotes the area of (n − 1)-dimensional sphere in Rn . The following statement is a direct consequence of Theorems 4.1 and 2.1 (ii). Corollary 4.1 Let M be a hyperconvex space and f : D → D  be an F AD-homeomorphism satisfying the inequality 0 < l(x, f ) ≤ L(x, f ) < ∞ for all x ∈ D. Suppose L(x, f ) ∈ L 1loc (D), then f ∈ ACn,1  (D). 4.2 Sobolev class W1,n and ring Q-homeomorphisms 1,n with the ring Here we connect the mappings of the standard Sobolev space Wloc homeomorphisms assuming integrability of the outer dilatation in degree n − 1. By the Lebesgue theorem on differentiation of nonnegative semiadditive locally finite set functions, the generalized Jacobian is well defined by

vg f (B(x, r )) a.e. r →0 vg B(x, r )

J (x, f ) = lim

See, e.g. [30, III.2.4]. Let K O (x, f ) denote the outer dilatation on M, which is defined by

K O (x, f ) =

⎧ ⎨

L n (x, f ) J (x, f ) ,

⎩ 1, ∞,

for J (x, f ) = 0, if L(x, f ) = 0, otherwise,

(16)

where L(x, f ) is given by (1). We define the inner dilatation K I (x, f ) of a mapping f on M by ⎧ J (x, f ) ⎨ l n (x, f ) K I (x, f ) = 1 ⎩ ∞

if J (x, f ) = 0, if l(x, f ) = 0, otherwise.

(17)

1,n The mappings of Sobolev class Wloc have many distinguished properties which closely related to the above mapping classes. For details and relations with the mentioned classes and around them see, e.g. [10,11,19,26,28]. Recall that a mapping f : D → D  between two Riemannian manifolds M and M 1, p (D ⊂ M and D  ⊂ M ) is of Sobolev class Wloc (D), if the coordinate functions of f have weak derivatives of the first order in local coordinates, and all these derivatives

30

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are integrable with the degree p. The definition of Wloc is correct due to its invariance with respect to the change of local coordinates. 1, p It is well known that a continuous mapping f of Wloc in Rn can be approximated by a sequence { f m } of smooth mappings such that f m → f converges uniformly, while its weak derivatives converge locally by norm in L p ; see, e.g. [38]. A similar result is true on smooth Riemannian manifolds, cf. [35, Theorem 6.10]. In the case p = n, there is a sequence { f m k } whose derivatives converge locally by norm in L 1 on a.a. locally rectifiable curves to f with respect to the measure of length paths; see [14]. Thus, similarly to [38, Theorem 28.2] a kind of Fuglede Theorem is provided in [1]. Proposition 4.1 Let D be a domain in a smooth Riemannian manifold M, and f : D → 1,n (D). Then M( ) = 0, where R be a continuous mapping of Sobolev class Wloc

stands for a family of all rectifiable curves in D, on which f fails to be locally absolutely continuous. Remark 4.4 Recall that homeomorphisms f between domains in Rn of Sobolev class 1,n are differentiable a.e. and possess the Lusin (N )-properties; see, e.g. [19]. If, Wloc 1,n in addition, K I (x, f ) ∈ L 1loc , then f −1 ∈ Wloc , and, therefore, f −1 has the same differential properties. The Lusin (N )-property for f −1 is equivalent to nonvanishing J (x, f ) a.e.; cf. [29]. Thus, in (16) and (17) the generalized Jacobian J (x, f ) can be replaced a.e. by J f (x) := | det f  (x)|, while l(x, f ) by l f (x) := inf |h|=1 | f  (x) · h|, and L(x, f ) by operator norm of Jacobi matrix  f  (x) := sup|h|=1 | f  (x) · h|. Here f  (x) denotes Jacobi martix of mapping f in any local coordinates on M and M . 1,n (D) with Theorem 4.3 Let f : D → D  be a homeomorphism of Sobolev class Wloc n−1 K O (x, f ) ∈ L loc (D). Then f is a ring Q-homeomorphism in D with Q(x) = n−1 (x, f ). C · KO

Proof Let be an arbitrary family of curves located in D, and ˜  be a family of all curves γ ∈ f ( ), on which f −1 is locally absolutely continuous. Then by Proposition 4.1 M( f ( )) = M( ˜  ). Let also A = A(x0 , ε, ε0 ), 0 < ε < ε0 < d(x0 ) := supx∈D d(x, x0 ), be an arbitrary geodesic ring in D, C ⊂ B(x0 , ε) ∩ D and C0 ⊂ D\B(x0 , ε0 ) be two continua, and ˜ = (C, C0 ; D) be a family of curves joining C and C0 in A. 1,n (D) is invariant by passing from chart to chart and due to Remark 4.4 Since f ∈ Wloc and Proposition 4.1, we can argue locally in D similarly to the proof of Lemma 2.4 in [15] and get dr /ds  ≤ L(y, f −1 ). Here r = d(x, x0 ). Without loss of generality, we may assume that η(r ) is a Borel function satisfying (6) and set ρ(y) ˜ = η(d( f −1 (y), f −1 (y0 )))L(y, f −1 ),

f −1 (y) = x, f −1 (y0 ) = x0 ,

for a.a. y ∈ D  , where f −1 is differentiable with L(y, f −1 ) = 0, and ρ(y) ˜ = ∞ otherwise at y ∈ f (A). Then, due to the invariance of the element of length, 



ε0

ρ˜ ds ≥ γ˜

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η(r ) dr ≥ 1 ε

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for all γ˜ ∈ ˜  , and, thus, ρ˜ ∈ adm ˜  . By to the assumptions of Theorem 4.3 and Remark 4.4, f and f −1 are differentiable a.e., possess the Lusin (N )-property and J (x, f ) = 0 a.e. in D. f ), where  f = ψ ◦ f ◦ϕ −1 , and (U , ϕ) and (V , ψ) Clearly, K I (x, f ) = K I (ϕ(x),  are normal coordinates of points x and f (x), respectively, such that f (U ) ⊂ V . This means that the quantity (17) is an invariant with respect to the changes of local coordinates. Hence, at every differentiability point x of f the outer dilatation K I (x, f ) can be calculated in the normal coordinates U (x0 ) in the same way as in the case of mappings between Euclidean domains; see [25, p. 5] and Remark 4.4. The similar arguments can be applied to the inner dilatation K I (x, f ). Now by the change of variables and due to the invariance of the volume element, we have  M(( f (C), f (C0 ); D  )) ≤ ρ˜ n dvg f (A)∩D 



ηn (d( f −1 (y), f −1 (y0 )))L n (y, f −1 ) dvg

= f (A)∩D 



ηn (d( f −1 (y), f −1 (y0 )))K I ( f −1 (y), f )J f −1 (y) dvg

= f (A)∩D 



=C·

K I (x, f )ηn (d(x, x0 )) dvg A∩D



n−1 KO (x, f )ηn (d(x, x0 )) dvg .

≤C· A∩D

Here we used the relations between the minimal and maximal stretching and Jacobians in the image and in the inverse one, i.e. J f −1 (y) = 1/J f ( f −1 (y)) and L(y, f −1 ) = 1/l f ( f −1 (y), f ) a.e. 

4.3 FAD and lower/ring/hyper Q-homeomorphisms The following results provide connections between the FAD-homeomorphisms with mappings whose definitions involve geometric approach only, namely, the lower Qhomeomorphisms, ring Q-homeomorphisms and hyper Q-homeomorphisms defined in Sect. 3. Theorem 4.4 Suppose that M is a hyperconvex space and f : D → D  is an FADk homeomorphism for some k = 1, . . . , n−1. Then f is both a lower Q-homeomorphism n−1 (·, f ). with Q = K O (·, f ) and a ring Q ∗ -homeomorphism with Q ∗ = c · K O (2) Proof By definition of FADk -homeomorphisms, f ∈ FMD and has (A(1) k ) and (Ak )properties. Applying Theorem 5.2 in [3] with the partition of unity subordinated to a locally finite atlas of the manifold, one concludes that f is lower Q-homeomorphism

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with Q = K O (x, f ). Here and in [3, Theorem 5.2], the hyperconvexity allows us to apply a Kirszbraun type theorem on Lipschitz extensions. Further, in view of Proposition 3.1 on connection between the ring and lower Qhomeomorphisms, we obtain that any FADk -homeomorphism f : D → D  is also a n−1 (x, f ), where K O (x, f ) ∈ L n−1 ring Q ∗ -homeomorphism with Q ∗ (x) = c · K O loc and the constant c is arbitrarily close to 1. 

Now combining Theorem 4.4 with [25, Theorem 10.1], we obtain the following statements. Corollary 4.2 Let M be a hyperconvex space. Suppose f : D → D  belongs to the class F ADk for some k = 1, ..., n − 1. Then f is a hyper Q-mapping in the dimension k, with Q(x, y) = (K I (x, f ), K I (y, f −1 )).

(18)

Corollary 4.3 Let M be a hyperconvex space. Every F AD mapping f is a hyper Q-mapping with Q given in (18). 4.4 Bi-Lipschitz mappings Here we discuss the behavior of mappings which by their definition should lie between L-bi-Lipschits and finitely bi-Lipschitz homeomorphisms. Suppose that f is a homeomorphism of M. For x ∈ M and r > 0 set l(x, r , f ) = inf d  ( f (x), f (y)) and L(x, r , f ) =

d(x,y)≥r

sup d  ( f (x), f (y)). d(x,y)≤r

The quantities defined in (1)–(2) can be treated as the maximal and minimal derivatives of f at x, respectively, and in terms of L(x, r , f ) and l(x, r , f ) they admit the following presentation L(x, f ) = lim sup r →0

L(x, r , f ) l(x, r , f ) , l(x, f ) = lim inf ; r →0 r r

cf. [8]. The following theorem shows the equivalence of two bi-Lipschitz representations. This statement is quatitative in the sense that the constants depend on each other and on the data associated with M only. Theorem 4.5 Let f : M → M be an L-bi-Lipschitz homeomorphism. Then it is equivalent that there exists a constant K ≥ 1 such that 1 ≤ l(x, f ) ≤ L(x, f ) ≤ K K holds for all x ∈ M. Reprinted from the journal

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E. Afanas’eva, A. Golberg

Proof The necessity is trivial and follows immediately from definition of L-biLipschitz mappings denoting d(x, y) = r and passing to r → 0. Thus, K = L. Now we prove the sufficiency. Assume that the inequality (19) holds. Then f is a K 2 -quasiconformal mapping by (3), and, therefore, f is absolutely continuous on almost all curves in M. This follows from the ACL-property of quasiconformal mappings. Choosing two arbitrary points x, y ∈ M, and a rectifiable curve γ joining x and y in M, on which f is absolutely continuous, one gets (γ ) ≤ C1 d(x, y)

(20)

where C1 ≥ 1 is a constant; cf. [8, Lemma 4.3]. Pick γ (s) as the path length parametrization of the curve γ , similarly to [36, Theorem 5.3], we obtain  ( f ◦ γ ) =

(γ )   ds ≤ L(γ (s), f ) ds ≤ L(x, f ) ds, 

f ◦γ

γ

0

Applying to the last integral the right-hand side in (19) and combining with (20), one gets d  ( f (x), f (y)) ≤ ( f ◦ γ ) ≤ K (γ ) ≤ K C1 d(x, y), and, therefore, d  ( f (x), f (y)) ≤ Ld(x, y). To prove the left-hand side of L-bi-Lipschitzness, we consider similarly to above a rectifiable curve β which joins f (x) and f (y) in M , and whose length satisfies (β) ≤ C2 d  ( f (x), f (y))

(21)

provided that f −1 is absolutely continuous on β. Let β(s) be the path length parametrization of β. Then the absolute continuity of f −1 on β allows us to conclude ( f

−1

 ◦ β) = f −1 ◦β

(β)    −1  ds ≤ L(β(s ), f ) ds = L(z, f −1 ) ds  . β

0

Taking in account the relation between the maximal and minimal derivatives of f and f −1 at the corresponding points, L(z, f −1 ) =

1 l( f −1 (z),

f)

,

we argue similarly to above and apply the left-hand side of (19) with (21), d(x, y) ≤ ( f −1 ◦ β) ≤ K (β) ≤ K C2 d  ( f (x), f (y)). 34

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This yields d(x, y)/L ≤ d  ( f (x), f (y)). Thus, the proof is completed.



4.5 Modulus criterion for bi-Lipschitzness Recall that the modulus of a spherical ring A in Rn is defined by  M(A) =

M( ) ωn−1



1 1−n

.

Due to the usual geometric (modular) definition of K -quasiconformal mappings (see, e.g. [36, Sec. 13.1]) that means that the modulus of curve families can change up to a constant factor, a homeomorphism is K -quasiconformal if and only if M(A) ≤ M( f (A)) ≤ K  M(A) K for all rings A, where K  = K 1/(n−1) ; cf. [36, Sec. 36.2]. In [31], it was established that for a proper subclass of quasiconformal mappings, namely L-bi-Lipschitz mappings of the whole Rn , the difference between the moduli of rings in the image and preimage is restricted by a constant from below and above. Thus, as a consequence following from Theorem 4.5, we obtain Corollary 4.4 Let f : M → M be a homeomorphism. Then the condition (19) holds for all x ∈ M if and only if there is a constant M such that |M( f (A)) − M(A)| ≤ M

(22)

for all geodesic rings A ⊂ M. This result generally cannot be quantitative, since for any conformal mapping M = 0 although K in (19) can be always chosen as bigger as possible (say even for a linear mapping). But if f fixes two points, M in (22) and K from (19) depend on each other.

5 Quasimöbius mappings In the general case, any quasisymmetric mapping is always quasimöbius, and both are quasiconformal. The converse implication between QM and QS has been studied in [37]. The result states that a QM-mapping f : X → Y is QS if the sets X and f (X ) are either both unbounded (with f (x) → ∞ as x → ∞) or both bounded. More precisely, see [37, Theorem 3.10]. A subset S of a metric space (X , d) is bounded if there exists r > 0 such that for all s and t in S, we have d(s, t) < r . (X , d) is a bounded metric space (or d is a bounded metric) if X is bounded as a subset of itself. Theorem 5.1 Let M and M be bounded connected Riemannian manifolds, z 1 , z 2 , z 3 ∈ M, and an embedding f : M → M be ω-quasimöbius such that d(z i , z j ) ≥ Reprinted from the journal

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diam M/λ, d  ( f (z i ), f (z j )) ≥ diam M /λ for i = j and λ > 0. Then f is an FAD-homeomorphism. Moreover, L(x, f ) is locally integrable with the degree n. Proof Since M and M are connected Riemannian manifolds, one can apply [7, Theorem 3.1, Ch. V] in order to pass to a metric space. We may also assume (without any loss of generality) that diam M = diam M = λ. Obviously, we can replace M by f (M), since f is an embedding; cf. [37, Theorem 3.12]. By the assumptions we can apply the mention above theorem from [37] on relations between ω-quasimöbius homeomorphisms and η-QS homeomorphisms and conclude that f is an η-QS homeomorphism. First, we show that f possesses the property of absolute continuity in measure which is equivalent to the Lusin (N )-property with respect to the n-dimensional Hausdorff measure, i.e. H n E = 0 for any compact set E ⊂ M implies H n f (E) = 0. To this end, we choose a countable covering of the set E by balls  Bi = B(xi , ri ), xi ∈ E, whose unit has sufficiently small measure, E ⊂ ∪i Bi , i (diam Bi )n < ε, for arbitrary ε > 0, i = 1, 2, . . . Applying a Vitali type covering theorem ( [17, Lemma 5.5]), one can find a countable disjoint subsequence of balls {  Bk , k = 1, 2, ...} for such covering Bk . For each k ≥ 1, due to the η-quasisymmetry of f , such that E ⊂ ∪k 5  B( f (x), r ) ⊆ f (B(x, r )) ⊆ f (k B(x, r )) ⊆ η(k)B( f (x), r ), cf. [5, Lemma 2.7]. Then H n f (E) ≤ H n f (∪k 5  Bk ) ≤



η(5)H n f (  Bk ) ≤



k

η(5)(diam f (  Bk ))n . (23)

k

Further, by the quasisymmetry and by inequality (7.27) in [17], we have the estimates   η(5)(diam f (  Bk ))n ≤ Cη [L δ (x, f )]n d H n (x) k

k

 = Cη  ≤ Cη

 Bk

[L δ (x, f )]n d H n (x)

Bk ∪k  ∪i Bi

(24)

[L δ (x, f )]n d H n (x).

Here Cη ≥ 1 and L δ (x, f ) = sup L(x, r , f )/r , 0 0 and L(x, r , f ) = quasisymmetry implies [L(x,

sup d  ( f (x), f (y)) for all x ∈ E. Recall that the

d(x,y)≤r f )]n ≤ C

η μ f (x),

μ f (x) = lim

r →0

where

H n f (B(x, r )) H n B(x, r ) 36

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stands for the volume derivative, that exists and is finite a.e. Moreover, μ f is locally integrable and satisfies the inequality G μ f (x) d H n (x) ≤ H n f (G) for any measurable subset G ⊂ M; see, e.g. [17]. Now combining (23) and (24) with the assumption that the n-dimensional Hausdorff measure of ∪i Bi is sufficiently small, we conclude that H n f (E) = 0. Let B = B(x0 , r ) be a normal neighborhood in M such that D ⊂ B, then f (B) is a normal neighborhood in M with D  ⊂ f (B). By Theorem 9.1 in [25], we deduce (1) that f has (Ak )-property on a.e. k-dimensional surface S in D. By n-regularity by Ahlfors of Riemannian manifolds (see, e.g. [1, Lemma 1]), we get applying (7.10) in [17] 

 [L(x, f )]n d H n (x) ≤ Cη

G

μ f (x) d H n (x), G

since L(x, f ) = limδ→0 L δ (x, f ). Recall that the Hausdorff measure H n f (G) < ∞ for any continuous mapping f (see, [13, 2.2.2] with [7, Theorem 3.1, Ch. V]) and G μ f (x) d H n (x) ≤ H n f (G) for any measurable subset G ⊂ M; see, e.g. [17]. Therefore, we have  [L(x, f )]n d H n (x) < ∞. G

Hence, L(x, f ) < ∞ a.e. and L(x, f ) is locally integrable with the degree n. Note that f −1 is an ω -quasimöbius homeomorphism by [37], and, therefore, f (2) possesses the Lusin (N −1 )-condition, and, consequently, (Ak )-property. Indeed, assuming H n f (E) = 0, we have H n E = H n f −1 ( f (E)) = 0 arguing similarly to above. Thus, one can conclude that f satisfies the Lusin (N −1 )-property. By The(2) orem 9.1 in [25], we get that f has (Ak )-property on a.e. k-dimensional surface S   in D ⊂ f (B). Arguing similarly for f −1 we obtain that L(y, f −1 ) = 1/l(x, f ) < ∞ a.e., which implies l(x, f ) > 0 a.e. Thus, f is an FAD-homeomorphism. The proof is complete. 

6 Applications to boundary correspondence results In this section we present some results related to the boundary extension for FADhomeomorphisms. Since any FAD-homeomorphism is an FMD-homeomorphism, the corresponding boundary behavior statements given in [3] hold for FAD-homeomorphisms. Reprinted from the journal

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Fig. 3 ∂ D is not weakly flat and ∂ D is weakly flat

6.1 Weak flatness and strong accessibility (weakly flatness and strong accessibility) are crucial for our studying boundary correspondences and provide valuable extensions of the well-known P1 and P2 -properties; see [36]. We say that the boundary of a domain D is weakly flat at a point x0 ∈ ∂ D, if for any number P > 0 and any neighborhood U of the point x0 there is a neighborhood V ⊂ U of x0 such that M((E, F; D)) ≥ P for all continua E and F in D intersecting ∂U and ∂ V . The boundary of a domain D is called weakly flat if this property holds at every point of the boundary; cf. [25] (Fig. 3). By [36, p. 53], the domain D has property P1 at its boundary point b, if E and F are connected subsets of D such that b ∈ E ∩ F, then M((E, F; D)) = ∞. It is clear that the weak flatness provides a deep extension of the property P1 . A wider condition than weak flatness on the boundary is a strong accessibility. We say that the boundary of a domain D is strongly accessible at a point x0 ∈ ∂ D, if for any neighborhood of U of the point x0 there are a compact set E ⊂ D, a neighborhood V ⊂ U of x0 and a number δ > 0 such that M((E, F; D)) ≥ δ for all continua F in D intersecting ∂U and ∂ V . Similarly to above, the boundary of a domain D is called strongly accessible if any boundary point of D is strongly accessible; cf. [25] (Fig. 4). This notion naturally extends the P2 -property at b ∈ ∂ D which has the following description (see [36, p. 54]). For any point point b1 ∈ ∂ D, b1 = b, there exist a 38

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Fig. 4 ∂ D is not strongly accessible and ∂ D is strongly accessible

compactum F ⊂ D and a constant δ > 0 such that M((E, F; D)) ≥ δ whenever E is a connected set in D such that E contains both b and b1 . 6.2 Boundary correspondence Further M is assumed to be a hyperconvex space. Some following results on the boundary behavior of FAD-homeomorphisms (applying Theorem 4.4) can be derived from the corresponding statements given for lower Q-homeomorphisms in [4] and for ring Q-homeomorphisms in [1]. For the reader convenience, either a sketch of the proof or a proof is provided here. We start with a sufficient condition on extensions to the boundaries for the inverse mappings to FADn−1 -homeomorphisms. Theorem 6.1 Let D be locally connected on ∂ D, D be compact, ∂ D  be weakly flat, and f be a homeomorphism of D onto D  of class F AD n−1 . Suppose that δ(x  0)

0

dr K O n−1 (x0 , r )

= ∞,

(25)

where 0 < δ(x0 ) < d(x0 ) = sup d(x, x0 ) such that B(x0 , δ(x0 )) is a normal neighborhood of the point x0 , and

x∈D

⎛ ⎜ K O n−1 (x0 , r ) = ⎝

⎞



⎟ n−1 KO (x) dAg ⎠

1 n−1

,

D(x0 ,r )

with D(x0 , r ) = {x ∈ D: d(x, x0 ) = r } = D ∩ S(x0 , r ). Then there is a continuous extension f −1 to D  . Reprinted from the journal

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Sketch of the proof By Theorem 4.4, f is a lower Q-homeomorphism with Q(x) = K O (x, f ) on B(x0 , δ(x0 )). Then arguing locally one can apply Lemma 9.6 from [25] and conclude that δ(x  0) σ

dr K O n−1 (x0 , r )

< ∞ ∀ σ ∈ (0, δ(x0 )).

Now we discuss similarly to the proof of Theorem 7.1 in [3] and reach the assertion of Theorem 6.1. 

Further we use the following notation of limit set C(x0 , f ) = {y ∈ M : y = lim f (xk ), xk → x0 , xk ∈ D}. k→∞

Lemma 6.1 Let D be locally connected at x0 ∈ ∂ D, D  be compact, and ∂ D  be strongly accessible at least at one point of the cluster set C(x0 , f ). Suppose that f : D → D  is an F ADn−1 -homeomorphism at x0 with K O ∈ L n−1 (D) such that  n−1 KO (x) · ψxn0 ,ε (d(x, x0 )) dvg = o(I xn0 (ε)) as ε → 0 D∩A

as for some sufficiently small ε0 ∈ (0, d(x0 )), where d(x0 ) = sup d(x, x0 ), A = x∈D

A(x0 , ε, ε0 ) and ψx0 ,ε (t) is a family of nonnegative Lebesgue measurable functions on (0, ∞) satisfying ε0 0 < I x0 (ε) : =

ψx0 ,ε (t) dt < ∞ ∀ ε ∈ (0, ε0 ), ε0 ∈ (0, d(x0 )). ε

Then f can be extended to the point x0 by continuity on M . Note that Lemma 6.1 provides a local condition for continuous boundary extension. The proof of the above lemma follows lines of the proof of [3, Theorem 7.2] given in a global form. Further, Lemma 6.1 allows us to formulate the following result. For details, see Theorem 7.3 in [3] proven also in a global case. Theorem 6.2 Let D be locally connected at x0 ∈ ∂ D, ∂ D  be strongly accessible, and D  be compact. Assume that the estimate (25) holds. Then any F AD n−1 -homeomorphism f : D → D  extends to the point x0 by continuity. The following lemma is a generalization of [32, Lemma 3.1] (given in Rn ) to Riemannian manifolds. 40

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Lemma 6.2 Let B0 = B(x0 , ε0 ) be a normal neighborhood of a point x0 in a Riemannian manifold M, K : B0 → (0, ∞) be a measurable function, and : [0, ∞] → [0, ∞] be an increasing convex function. Then ε0 0

∞ dτ c ≥ 1 1 n r k p (r ) τ [−1 (τ )] p eM dr

∀ p ∈ (0, ∞),

0

where M0 is the average of  ◦ K over B0 , k(r ) is the mean value of K (x) over a geodesic sphere S(x0 , r ), r ∈ (0, ε0 ), and c is a constant sufficiently close to 1 for small ε0 . Theorem 6.3 Let D be locally connected at x0 ∈ ∂ D, ∂ D  be strongly accessible, and D  be compact. Suppose that f : D → D  is an F AD n−1 -homeomorphism with K O ∈ L n−1 (D),  n−1 (K O (x)) dvg < ∞

(26)

D

for a non-decreasing convex function : [0, ∞] → [0, ∞] such that ∞

dτ 1

δ

τ [−1 (τ )] n−1

= ∞ as δ > (0).

(27)

Then f extends to the point x0 by continuity. Arguing similarly to [32, Theorem 3.1], we apply Lemma 6.2, since conditions (26) and (27) yield (25). This allows us to reach the assertion of Theorem 6.3 as a direct conclusion of Theorem 6.2. Corollary 6.1 The conclusion of Theorem 6.3 holds if for some α > 0, 

n−1

eα K O

(x)

dvg < ∞.

D

Data availability The manuscript has no associated data.

Declarations Conflict of interest The authors declare that there is no conflict of interest.

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References 1. Afanas’eva, E.S.: Boundary behavior of ring Q-homeomorphisms on Riemannian manifolds. Ukr. Math. J. 63(10), 1479–1493 (2012) 2. Afanas’eva, E., Golberg, A.: Finitely bi-Lipschitz homeomorphisms between Finsler manifolds. Anal. Math. Phys. 10(4), paper no. 48, 16 (2020) 3. Afanas’eva, E., Golberg, A.: Homeomorphisms of finite metric distortion between Riemannian manifolds. Comput. Methods Funct. Theory (2022). https://doi.org/10.1007/s40315-021-00431-3 4. Afanas’eva, E.S., Ryazanov, V.I., Salimov, R.R.: On mappings in Orlicz–Sobolev classes on Riemannian manifolds (Russian). Ukr. Mat. Vis. 8(3), 319–342, 461 (2011). (Translation in J. Math. Sci. (N.Y.) 181 (2012), no. 1, 1–17) 5. Bishop, C.J., Hakobyan, H., Williams, M.: Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space. Geom. Funct. Anal. 26, 379–421 (2016) 1,n 6. Bongiorno, D.: On the problem of regularity in the Sobolev space Wloc . Topol. Appl. 156(18), 2986– 2995 (2009) 7. Boothby, W.M.: An Introduction to Differentiable Manifolds and Riemannian Geometry. Pure and Applied Mathematics, vol. 120, 2nd edn. Academic Press, Inc., Orlando (1986) 8. Chen, K.Y., Fang, A.N.: Bi-Lipschitz maps in Q-regular Loewner spaces. Acta Math. Sin. (Engl. Ser.) 24(9), 1555 (2008) 9. Cristea, M.: The limit mapping of generalized ring homeomorphisms. Complex Var. Elliptic Equ. 61(5), 608–622 (2016) 10. Cristea, M.: A generalization of the univalence on the boundary theorem with applications to Sobolev mappings. Complex Anal. Oper. Theory 11(8), 1789–1799 (2017) 11. Cristea, M.: A note on the global injectivity of some light Sobolev mappings. J. Math. Pures Appl. (9) 128, 213–224 (2019) 12. Espinola, R., Nicolae, A.: Continuous selections of Lipschitz extensions in metric spaces. Rev. Mat. Comput. 28(3), 741–759 (2015) 13. Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, Band 153. Springer, New York (1969) 14. Fuglede, B.: Extremal length and functional completion. Acta Math. 98, 171–219 (1957) 15. Gutlyanski˘ı, V.Y., Golberg, A.: On Lipschitz continuity of quasiconformal mappings in space. J. Anal. Math. 109, 233–251 (2009) 16. Heinonen, J.: Lectures on Analysis on Metric Spaces. Universitext. Springer, New York (2001) 17. Heinonen, J., Koskela, P.: Quasiconformal maps in metric spaces with controlled geometry. Acta Math. 181(1), 1–61 (1998) 18. Hencl, S.: On the notions of absolute continuity for functions of several variables. Fund. Math. 173(2), 175–189 (2002) 19. Hencl, S., Koskela, P.: Lectures on Mappings of Finite Distortion. Lecture Notes in Mathematics, vol. 2096. Springer, Cham (2014) 20. Kovtonyuk, D., Petkov, I., Ryazanov, V.: Prime ends in the theory of mappings with finite area distortion in the plane. Filomat 31(5), 1349–1366 (2017) 21. Kovtonyuk, D., Ryazanov, V.: On the theory of mappings with finite area distortion. J. Anal. Math. 104, 291–306 (2008) 22. Lee, J.M.: Riemannian Manifolds: An Introduction to Curvature. Graduate Texts in Mathematics, vol. 176. Springer, New York (1997) 23. Malý, J.: Absolutely continuous functions of several variables. J. Math. Anal. Appl. 231(2), 59–61 (1999) 24. Martio, O., Rickman, S., Väisälä, J.: Definitions for quasiregular mappings. Ann. Acad. Sci. Fenn. Ser. A I No. 448, 40 (1969) 25. Martio, O., Ryazanov, V., Srebro, U., Yakubov, E.: Moduli in Modern Mapping Theory. Springer Monographs in Mathematics. Springer, New York (2009) 26. Maz’ja, V.G.: Sobolev Spaces. Translated from the Russian by T. O. Shaposhnikova. Springer Series in Soviet Mathematics. Springer, Berlin (1985) 27. Maz’ya, V.: Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002). Contemporary Mathematics, vol. 338, pp. 307–340. American Mathematical Society, Providence (2003)

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Analysis and Mathematical Physics (2021) 11:84 https://doi.org/10.1007/s13324-021-00511-6

On fractional Orlicz–Sobolev spaces Angela Alberico1 · Andrea Cianchi2

· Luboš Pick3 · Lenka Slavíková3

Received: 4 October 2020 / Accepted: 19 February 2021 / Published online: 23 March 2021 © The Author(s) 2021

Abstract Some recent results on the theory of fractional Orlicz–Sobolev spaces are surveyed. They concern Sobolev type embeddings for these spaces with an optimal Orlicz target, related Hardy type inequalities, and criteria for compact embeddings. The limits of these spaces when the smoothness parameter s ∈ (0, 1) tends to either of the endpoints of its range are also discussed. This note is based on recent papers of ours, where additional material and proofs can be found. Keywords Fractional Orlicz–Sobolev spaces · Sobolev embeddings · Compact embeddings · Limits of fractional seminorms · Orlicz spaces · Rearrangement-invariant spaces Mathematics Subject Classification 46E35 · 46E30

Dedicated to Vladimir Maz’ya with esteem and admiration.

B

Andrea Cianchi [email protected] Angela Alberico [email protected] Luboš Pick [email protected] Lenka Slavíková [email protected]

1

Istituto per le Applicazioni del Calcolo “M. Picone”, Consiglio Nazionale delle Ricerche, Via Pietro Castellino 111, 80131 Napoli, Italy

2

Dipartimento di Matematica e Informatica “U. Dini”, Università di Firenze, Viale Morgagni 67/a, 50134 Firenze, Italy

3

Department of Mathematical Analysis, Faculty of Mathematics and Physics, Charles University, Sokolovská 83, 186 75 Praha 8, Czech Republic

A previous version of this chapter was published Open Access under a Creative Commons Attribution 4.0 International License at http://link.springer.com/10.1007/s13324-021-00511-6. Chapter 3 was originally published as Alberico, A., Cianchi, A., Pick, L. & Slavíková, L. Analysis and Mathematical Physics (2021) 11:84. https://doi.org/10.1007/s13324-021-00511-6. Reprinted from the journal

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1 Introduction One of the available notions of Sobolev spaces of fractional order calls into play the Gagliardo–Slobodeckij seminorm. Given an open set  ⊂ Rn , with n ∈ N, and numbers s ∈ (0, 1) and p ∈ [1, ∞), this seminorm will be denoted by | · |s, p, , and is defined as    |u|s, p, =

 

|u(x) − u(y)| |x − y|s

p

dx dy |x − y|n

1

p

(1.1)

for a measurable function u :  → R. The fractional Sobolev space W s, p () is defined as the Banach space of those functions u for which the norm uW s, p () = u L p () + |u|s, p,

(1.2)

is finite. Standard properties of the spaces W s, p () are classical. The last two decades have witnessed an increasing number of investigations on these spaces because of their use in the analysis of nonlocal elliptic and parabolic equations, whose study has received an enormous impulse in the same period – see e.g. [7–22,24–27,31,34,35,38– 40,43–54,56,57,59,60,64]. The aim of this note is to survey a few recent results, contained in [1–4], on some aspects of fractional Orlicz–Sobolev spaces. They constitute an extension of the spaces W s, p (), in that the role of the power function t p is performed by a more general finite-valued Young function A(t), namely a convex function from [0, ∞) into [0, ∞), vanishing at 0. The fractional Orlicz–Sobolev space, of order s ∈ (0, 1), associated with a Young function A, will be denoted by W s,A (), and is built upon the Luxemburg type seminorm | · |s,A, given by |u|s,A,

      dx dy |u(x) − u(y)| = inf λ > 0 : A ≤1 λ|x − y|s |x − y|n  

(1.3)

for a measurable function u :  → R. The norm of a function u in W s,A () is accordingly defined as uW s,A () = u L A () + |u|s,A, ,

(1.4)

where u L A () stands for the Luxemburg norm in the Orlicz space L A (). Definitions (1.3) and (1.4) have been introduced in [37], where some basic properties of the space W s,A () are analyzed under the 2 and ∇2 conditions on A. Plainly, these definitions recover (1.1) and (1.2) when A(t) = t p for some p ∈ [1, ∞). Sobolev embeddings for the space W s, p () have been long known. In particular, if s ∈ (0, 1) and 1 ≤ p < ns , then there exists a constant C such that u

np

L n−sp (Rn )

≤ C|u|s, p,Rn 46

(1.5)

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for every measurable function u : Rn → R decaying to 0 near infinity. A companion result holds if Rn is replaced by any bounded open set  with a sufficiently regular boundary ∂, for any function u ∈ W s, p (), provided that the seminorm |u|s, p,Rn is replaced by the norm uW s, p () . Sharp extensions of these Sobolev type inequalities and ensuing embeddings to the spaces W s,A () are presented in Sect. 3. For instance, the optimal Orlicz target space L B (Rn ) in the inequality u L B (Rn ) ≤ C|u|s,A,Rn , for some constant C and every measurable function u : Rn → R decaying to 0 near infinity, is exhibited. Compact embeddings are also characterized. Here, we shall limit ourselves to consider target spaces of Orlicz type. However, inequalities and embeddings involving even stronger rearrangement-invariant norms are available. For these results we refer to [1,4], where proofs of the material collected in this paper can also be found. Let us add that in those papers optimal embeddings for higher-order fractional spaces W s,A () associated with any s ∈ (0, n)\N are established as well. A second issue that will be addressed concerns the limits as s → 1− and s → 0+ of the space W s,A (Rn ). It is well known that setting s = 1 in the definition of the space W s, p (Rn ) does not recover the first-order Sobolev space W 1, p (Rn ). Moreover, the Lebesgue space L p (Rn ) cannot be obtained on choosing s = 0 in the definition of W s,A (Rn ). Still, the seminorm ∇u L p (Rn ) and the norm u L p (Rn ) of a function are reproduced as limits as s → 1− and s → 0+ , respectively, of the seminorm |u|s, p,Rn , provided that the latter is suitably normalized by a multiplicative factor depending on s, p and n. Specifically, a version in the whole of Rn of a result by Bourgain-Brezis-Mironescu [9,10] tells us that, if p ∈ [1, ∞), then  lim (1 − s)

s→1−

 Rn

Rn



|u(x) − u(y)| |x − y|s

p

dx dy = K ( p, n) |x − y|n

 Rn

|∇u(x)| p d x (1.6)

for every function u ∈ W 1, p (Rn ), where  1 |θ · e| p dHn−1 . K ( p, n) = p Sn−1

(1.7)

Here, Sn−1 denotes the (n − 1)-dimensional unit sphere in Rn , Hn−1 denotes the (n − 1)-dimensional Hausdorff measure, e is any point on Sn−1 , and the dot “ · ” stands for scalar product in Rn . Conversely, if p ∈ (1, ∞), u ∈ L p (Rn ) and the limit (or even the liminf) on the left-hand side of (1.6) is finite, then u ∈ W 1, p (Rn ). The case when p = 1 is excluded from the latter result, but has a counterpart with W 1,1 (Rn ) replaced by BV (Rn ), the space of functions of bounded variation in Rn . A slight variant of these facts is proved in [62]. In the precise form appearing above, they follow as special cases of results of [3]. A version of Eq. (1.6) with Rn replaced by a bounded regular domain can be found in [32]. Reprinted from the journal

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The limit as s → 0+ is the subject of a theorem by Maz’ya–Shaposhnikova [50], which ensures that  lim s

s→0+



 Rn

Rn

|u(x) − u(y)| |x − y|s

p

dx dy 2 nωn = |x − y|n p

 Rn

|u(x)| p d x

(1.8)

for each p ∈ [1, ∞), and for every function u decaying to 0 near infinity and making the double integral finite for some s ∈ (0, 1). Here, ωn denotes the Lebesgue measure of the unit ball in Rn . Equation (1.8) has to be interpreted in the sense that u ∈ L p (Rn ) if and only if the limit on the left-hand side if finite, and that, in the latter case, the equality holds. Section 4 is devoted to counterparts, established in [3] and [2], of these results in the Orlicz framework. Namely, it deals with the limits  lim (1 − s)

s→1−



 Rn

Rn

A

|u(x) − u(y)| |x − y|s



dx dy , |x − y|n

(1.9)

and  lim s

s→0+





Rn

Rn

A

|u(x) − u(y)| |x − y|s



dx dy . |x − y|n

(1.10)

Interestingly, the conclusions about these limits share some features with those in (1.6) and (1.8), but also present some diversities. In particular, as shown by counterexamples, certain results can only hold under the additional 2 -condition on A, or are affected by some restrictions in the general case.

2 Function spaces A function A : [0, ∞) → [0, ∞] is called a Young function if it has the form  A(t) =

t

a(τ )dτ

for t ≥ 0,

0

for some non-decreasing, left-continuous function a : [0, ∞) → [0, ∞] which is neither identically equal to 0 nor to ∞. Clearly, any convex (non trivial) function from [0, ∞) into [0, ∞], which is left-continuous and vanishes at 0, is a Young function. A Young function A is said to dominate another Young function B globally if there exists a positive constant C such that B(t) ≤ A(Ct)

for t ≥ 0 .

(2.1)

The function A is said to dominate B near infinity if there exists t0 > 0 such that (2.1) holds for t ≥ t0 . 48

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The function B is said to grow essentially more slowly near infinity than A if lim

t→∞

B(λt) =0 A(t)

(2.2)

for every λ > 0. Note that condition (2.2) is equivalent to lim

t→∞

A−1 (t) = 0. B −1 (t)

(2.3)

A Young function A is said to satisfy the 2 -condition – briefly A ∈ 2 – globally if there exists a positive constant C such that A(2t) ≤ C A(t)

(2.4)

for t ≥ 0. If A is finite-valued and there exists t0 > 0 such that inequality (2.4) holds for t ≥ t0 , then we say that A satisfies the 2 -condition near infinity. Let  be a measurable subset of Rn , with n ≥ 1, having Lebesgue measure ||. Set M() = {u :  → R : u is measurable}, and M+ () = {u ∈ M() : u ≥ 0} . The notation Md () is employed for the subset of M() of those functions u that decay near infinity, according to the following definition: Md () = {u ∈ M() : |{|u| > t}| < ∞ for every t > 0} . Plainly, Md () = M() if || < ∞. The Orlicz space L A (), associated with a Young function A, is the Banach space of those functions u ∈ M() for which the Luxemburg norm      |u(x)| dx ≤ 1 u L A () = inf λ > 0 : A λ  is finite. In particular, L A () = L p () if A(t) = t p for some p ∈ [1, ∞), and L A () = L ∞ () if A(t) = 0 for t ∈ [0, 1] and A(t) = ∞ for t ∈ (1, ∞). If A dominates B globally, then u L B () ≤ Cu L A ()

(2.5)

for every u ∈ L A (), where C is the same constant as in (2.1). If || < ∞, and A dominates B near infinity, then inequality (2.5) continues to hold for some constant C = C(A, B, t0 , ||). Reprinted from the journal

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The alternative notation A(L)() will also be employed, in the place of L A (), to denote the Orlicz space associated with a Young function equivalent to A. The space E A () is defined as      |u(x)| d x < ∞ for every λ > 0 . E A () = u ∈ M() : A λ  If A is finite-valued, then the space E A () agrees with the closure in L A () of the space of bounded functions with bounded support in . Trivially, E A () ⊂ L A () . This inclusion holds as equality if and only if either || < ∞ and A ∈ 2 near infinity, or || = ∞ and A ∈ 2 globally. Assume now that  is an open subset of Rn . We denote by V 1,A () the homogeneous Orlicz–Sobolev space given by   1,1 V 1,A () = u ∈ Wloc () : |∇u| ∈ L A () . Here, ∇u denotes the gradient of u. The notation W 1,A () is adopted for the classical Orlicz–Sobolev space defined by   W 1,A () = u ∈ L A () : |∇u| ∈ L A () . The space W 1,A () is a Banach space equipped with the norm uW 1,A () = u L A () + ∇u L A () . By W 1 E A () we denote the space obtained on replacing L A () with E A () in the definition of W 1,A (). The space of functions of bounded variation on  is denoted by BV (). It consists of all functions in L 1 () whose distributional gradient is a vector-valued Radon measure Du with finite total variation Du() on . The space BV () is a Banach space, endowed with the norm defined as u BV () = u L 1 () + Du() for u ∈ BV (). Given a function u ∈ BV (), we denote by ∇u the absolutely continuous part of Du with respect to the Lebesgue measure, and by D s u its singular part. One has that  Du() =



|∇u| d x + D s u(),

where D s u() stands for the total variation of the measure D s u over . 50

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More generally, assume that A is a Young function with a linear growth near infinity, in the sense that lim

t→∞

A(t) < ∞. t

(2.6)

Then a functional J A, associated with A can be defined on BV () as  J A, (u) =



A (|∇u|) d x + a ∞ D s u()

(2.7)

for u ∈ BV (), where a ∞ = lim

t→∞

A(t) . t

(2.8)

The functional J A, agrees on BV () with the relaxed functional of  

A(|∇u|) d x

on L 1 () with respect to convergence in L 1loc (), which is defined as  inf

 lim inf m→∞



 A(|∇u m |) d x : {u m } ⊂ C 1 (), u m → u in L 1loc () .

One has that the functional J A, is lower semicontinuous in BV () with respect to convergence in L 1loc (). Moreover, for every function u ∈ BV (), there exists a sequence {u m } ⊂ C 1 () such that  u m → u in

L 1loc ()

and J A, (u) = lim

m→∞ 

A(|∇u m |) d x.

The homogeneous fractional Orlicz–Sobolev space V s,A () is defined as V s,A () = u ∈ M() : |u|s,A, < ∞} , where | · |s,A, is the seminorm given by (1.3). The subspace of those functions in V s,A () that decay near infinity is denoted by Vds,A (). Namely Vds,A () = V s,A () ∩ Md (). If || < ∞ and s ∈ (0, 1), we also define the space   V⊥s,A () = u ∈ V s,A () : u  = 0 , Reprinted from the journal

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where u =

1 ||

 u dx ,



the mean value of u over . The fractional-order Orlicz–Sobolev space W s,A () is defined as   W s,A () = u ∈ L A () : u ∈ V s,A () , and is a Banach space equipped with the norm given by (1.4). Clearly, W s,A () → Vds,A (), and, as a consequence of Proposition 1, Sect. 3, W s,A () = Vds,A () if  is bounded. We conclude by mentioning a fractional-order Pólya–Szeg˝o principle, which implies the decrease of the fractional Orlicz–Sobolev seminorm under symmetric rearrangement of functions u. Recall that the symmetric rearrangement u  of a function u ∈ Md (Rn ) is defined as the radially decreasing function about 0 which is equidistributed with u. Theorem 2.1 (Fractional Pólya–Szeg˝o principle) Let s ∈ (0, 1) and let A be a Young function. Assume that u ∈ Md (Rn ). Then

       |u  (x) − u  (y)| dx dy dx dy |u(x) − u(y)| A ≥ A . s n s n n n n |x − y| |x − y| |x − y| |x − y|n R R R R (2.9) In the case when A is a power, inequality (2.9) can be traced back to [5,6]. The result for Young functions A satisfying the 2 -condition and functions u ∈ W s,A (Rn ) is proved in [33]. The general version stated in Theorem 2.1 can be found in [1]. An earlier related contribution, dealing with functions of one-variable, is [42].

3 Sobolev type inequalities Our first theorem provides us with the optimal – i.e. smallest possible – Orlicz target space in the Sobolev embedding for the space Vds,A (Rn ). Such an optimal space is built upon the Young function A ns associated with A, n and s as follows. Let s ∈ (0, 1) and let A be a Young function such that 

∞

t A(t)



s n−s

dt = ∞

(3.1)

dt < ∞.

(3.2)

and   0

t A(t)



s n−s

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Then, A ns is given by A ns (t) = A(H −1 (t)) for t ≥ 0,

(3.3)

where the function H : [0, ∞) → [0, ∞) obeys H (t) =

 t  0

τ A(τ )



 n−s

s n−s

n

dτ

for t ≥ 0.

Theorem 3.1 (Optimal Orlicz target space) Let s ∈ (0, 1). Assume that A is a Young function satisfying conditions (3.1) and (3.2), and let A ns be the Young function defined as in (3.3). Then Vds,A (Rn ) → L

An s

(Rn ),

(3.4)

and there exists a constant C = C(n, s) such that u

L

An s

(Rn )

≤ C|u|s,A,Rn

for every function u ∈ Vds,A (Rn ). Moreover, L inequality (3.5) among all Orlicz spaces.

An s

(3.5)

(Rn ) is the optimal target space in

Remark 1 Assumption (3.2) on the Young function A cannot be dispensed with in Theorem 3.1. Actually, one can show that it is necessary for an embedding of the space Vds,A (Rn ) to hold into any rearrangement-invariant space. Assumption (3.1) amounts to requiring that A has a subcritical growth with respect to the smoothness parameter s. It generalizes the condition p < ns required for the classical inequality (1.5). Remark 2 The fractional Orlicz–Sobolev inequality (3.5) precisely matches the integer-order inequality established in [29] (see also [28] for an alternative form). Indeed, setting s = 1 in formula (3.3) for the function A ns recovers the Young function which defines the optimal Orlicz target space in the Orlicz–Sobolev inequality for W 1,A (Rn ). We now present an application of Theorem 3.1 to a family of Young functions whose behaviour near zero and near infinity is of power-logarithmic type. Although quite simple, these model Young functions enable us to recover results available in the literature and to exhibit genuinely new inequalities. Example 1 Consider a Young function A such that A(t) is equivalent to

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(3.6)

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where either p0 > 1 and α0 ∈ R, or p0 = 1 and α0 ≤ 0, and either p > 1 and α ∈ R, or p = 1 and α ≥ 0. Here, equivalence is meant in the sense of Young functions. Let s ∈ (0, 1). The function A satisfies assumption (3.1) if either 1 ≤ p
0. (ii) The embedding B Vds,A (Rn ) → L loc (Rn )

(3.16)

is compact. (iii) The embedding W s,A () → L B () is compact for every bounded Lipschitz domain  in Rn . The assertion that embedding (3.16) is compact means that every bounded sequence in Vds,A (Rn ) has a subsequence whose restriction to E converges in L B (E) for every bounded measurable set E in Rn . Let us notice that the equivalence of properties (i) and (ii) is not explicitly mentioned in [4]. Its proof follows, modulo minor variants, along the same lines as that of the equivalence of (i) and (iii). Example 4 Assume that  is a bounded Lipschitz domain in Rn . Let s ∈ (0, 1) and let A be a Young function as in (3.6), (3.7) and (3.8). From Theorem 3.5 and property (2.3) one infers that the embedding W s,A () → L B () is compact if and only if B is a Young function fulfilling ⎧ ⎪ ⎪ lim ⎪ ⎪ ⎨ t→∞ limt→∞ ⎪ ⎪ ⎪ ⎪ ⎩lim t→∞

t

n−sp np

(log t) B −1 (t)

− αp

=0

if 1 ≤ p
0. 60

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Remark 3 Let us emphasize that, unlike Theorem 4.1, the 2 -condition is not required in Theorem 4.3, at the expense of replacing u by u/λ for sufficiently large λ > 0. This is consistent with the fact that, if A does not satisfy member this condition,  d x, and hence ship of ∇u in the Orlicz space L A (Rn ) only ensures that Rn A |∇u| λ   |∇u|  d x, is finite for sufficiently large λ. However, under the  A -condition on 2 Rn ◦ λ 1,A n 1,A n A, one has that W (R ) = W (R ), and hence Eq. (4.4) holds for every λ > 0, including λ = 1. In the framework of Orlicz spaces associated with a Young function A, an analogue of the distinction between p = 1 and p ∈ (1, ∞) for powers is properly formulated in terms of the limit at infinity and/or at 0 of the (non-decreasing) function A(t)/t. In particular, a converse to Theorem 4.3 holds under the superlinear growth condition on A near infinity lim

t→∞

A(t) = ∞, t

(4.5)

A(t) = 0. t

(4.6)

and the sublinear decay condition at 0 lim

t→0+

Plainly, if A(t) = t p , either of conditions (4.5) and (4.6) is equivalent to requiring that p > 1. Theorem 4.4 Let A be a finite-valued Young function. Assume that A fulfills conditions (4.5) and (4.6). If u ∈ L A (Rn ) is such that  lim inf (1 − s) s→1−



 Rn

Rn

A

|u(x) − u(y)| λ|x − y|s



dx dy 0, then u ∈ W 1,A (Rn ). In the case when A has a linear growth near infinity or near 0, Theorems 4.3 and 4.4, respectively, have counterparts in the framework of functions of bounded variation. Assume that A is a Young function for which equation (4.5) fails, and hence condition (2.6) holds. Since the function A◦ given by (4.3) is equivalent to A, Eq. (2.6) also holds if A is replaced by A◦ . Let a◦∞ be the number defined as in (2.8), with A replaced by A◦ , namely a◦∞ = lim

t→∞

A◦ (t) . t

The following result tells us that, under (2.6), if u ∈ BV (Rn ) then the limit in (4.4) the relaxed functional of equals the functional J A◦ ,Rn (u) defined as in (2.7),1 namely n ). A (|∇u|) d x with respect to convergence in L (R loc Rn ◦ Reprinted from the journal

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Theorem 4.5 Let A be a Young function fulfilling condition (2.6). Assume that u ∈ BV (Rn ). Then,  lim (1 − s)

s→1−



 Rn

Rn

A

|u(x) − u(y)| |x − y|s



dx dy = |x − y|n

 Rn

A◦ (|∇u|) d x + a◦∞ D s u(Rn ).

Suppose now that condition (4.6) does not hold, namely lim

t→0+

A(t) > 0. t

(4.8)

From Eq. (4.7) one can conclude that u ∈ BV (Rn ). Theorem 4.6 Let A be a Young function fulfilling condition (4.8). Assume that u ∈ L 1 (Rn ) is such that  lim inf (1 − s) s→1−

 Rn

Rn



|u(x) − u(y)| A λ|x − y|s



dx dy 0. Then u ∈ BV (Rn ). Funding Open access funding provided by Universitá degli Studi di Firenze within the CRUI-CARE Agreement. This research was partly funded by: [1.] Research Project 201758MTR2 of the Italian Ministry of Education, University and Research (MIUR) Prin 2017 “Direct and inverse problems for partial differential equations: theoretical aspects and applications”; [2.] GNAMPA of the Italian INdAM—National Institute of High Mathematics (Grant number not available); [3.] Grant P201-18-00580S of the Czech Science Foundation.

Declaration Conflict of interest The authors declare that they have no conflict of interest. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

References 1. Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: Fractional Orlicz-Sobolev embeddings. J. Math. Pures Appl., to appear 2. Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: On the limit as s → 0+ of fractional Orlicz-Sobolev spaces. J. Fourier Anal. Appl. 26, no. 6, Paper No. 80, 19 pp (2020) 3. Alberico, A., Cianchi, A., Pick, L., Slavíková, L.: On the limit as s → 1− of possibly non-separable fractional Orlicz-Sobolev spaces. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31, 879–899 (2020)

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A. Alberico et al. 32. Dávila, J.: On an open question about functions of bounded variation. Calc. Var. Partial Differ. Equ. 15(4), 519–527 (2002) 33. De Nápoli, P., Fernández Bonder, J., Salort, A. M.: A Pólya–Szeg˝o principle for general fractional Orlicz–Sobolev spaces. Complex Var. Elliptic Equ. 1–23, (2020) 34. Dyda, B., Frank, R.: Fractional Hardy–Sobolev–Maz’ya inequality for domains. Studia Math. 208, 151–166 (2012) 35. Dyda, B., Vähäkangas, A.V.: Characterizations for fractional Hardy inequality. Adv. Calc. Var. 8, 173–182 (2015) 36. Edmunds, D.E., Gurka, P., Opic, B.: Double exponential integrability of convolution operators in generalized Lorentz–Zygmund spaces. Indiana Univ. Math. J. 44, 19–43 (1995) 37. Fernández-Bonder, J., Salort, A.M.: Fractional order Orlicz–Sobolev spaces. J. Funct. Anal. 227, 333– 367 (2019) 38. Figalli, A., Fusco, N., Maggi, F., Millot, V., Morini, M.: Isoperimetry and stability properties of balls with respect to nonlocal energies. Commun. Math. Phys. 336, 441–507 (2015) 39. Filippas, S., Moschini, L., Tertikas, A.: Sharp trace Hardy-Sobolev-Maz’ya inequalities and the fractional Laplacian. Arch. Ration. Mech. Anal. 208, 109–161 (2013) 40. Frank, R., Jin, T., Xiong. J.: Minimizers for the fractional Sobolev inequality on domains. Calc. Var. Partial Differ. Equ. 57 (2018) Art. 43, 31 pp 41. Fusco, N., Lions, P.-L., Sbordone, C.: Sobolev imbedding theorems in borderline cases. Proc. Am. Math. Soc. 124, 561–565 (1996) 42. Garsia, A.M., Rodemich, E.: Monotonicity of certain functionals under rearrangement. Ann. Inst. Fourier 24, 67–116 (1974) 43. Heuer, N.: On the equivalence of fractional-order Sobolev semi-norms. J. Math. Anal. Appl. 417, 505–518 (2014) 44. Kuusi, T., Mingione, G., Sire, Y.: Nonlocal self-improving properties. Anal. PDE 8, 57–114 (2015) 45. Ludwig, M.: Anisotropic fractional Sobolev norms. Adv. Math. 252, 150–157 (2014) 46. Mallick, A.: Extremals for fractional order Hardy–Sobolev–Maz’ya inequality. Calc. Var. Partial Differ. Equ. 58 (2019) Art. 45, 37 pp 47. Marano, S., Mosconi, S.J.N.: Asymptotics for optimizers of the fractional Hardy–Sobolev inequality. Commun. Contemp. Math. 21, 1850028 (2019). 33 pp 48. Maz’ya, V.G.: Sobolev Spaces with Applications to Elliptic Partial Differential Equations. Springer, Berlin (2011) 49. Maz’ya, V.G., Shaposhnikova, T.: On the Brezis and Mironescu conjecture concerning a Gagliardo– Nirenberg inequality for fractional Sobolev norms. J. Math. Pures Appl. 81, 877–884 (2002) 50. Maz’ya, V.G., Shaposhnikova, T.: On the Bourgain, Brézis and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195, 230–238 (2002) 51. Maz’ya, V.G., Shaposhnikova, T.: Theory of Sobolev Multipliers, with Applications to Differential and Integral Operators. Springer, Berlin (2009) 52. Musina, R., Nazarov, A.I.: A note on truncations in fractional Sobolev spaces. Bull. Math. Sci. 9, 1950001 (2019). 7 pp 53. Palatucci, G., Pisante, A.: Improved Sobolev embeddings, profile decomposition, and concentrationcompactness for fractional Sobolev spaces. Calc. Var. Partial Differ. Equ. 50, 799–829 (2014) 54. Parini, E., Ruf, B.: On the Moser–Trudinger inequality in fractional Sobolev–Slobodeckij spaces. J. Anal. Math. 138, 281–300 (2019) 55. Pohozhaev, S.I.: On the imbedding Sobolev theorem for pl = n. Doklady Conference Section Math. Moscow Power Inst. 165, 158–170 (1965). (Russian) 56. Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213, 587–628 (2014) 57. Ros-Oton, X., Serra, J.: Boundary regularity for fully nonlinear integro-differential equations. Duke Math. J. 165, 2079–2154 (2016) 58. Salort, A.M.: Hardy inequalities in fractional Orlicz-Sobolev spaces. Publ. Mat., to appear 59. Seeger, A., Trebels, W.: Embeddings for spaces of Lorentz–Sobolev type. Math. Ann. 373, 1017–1056 (2019) 60. Tzirakis, K.: Sharp trace Hardy–Sobolev inequalities and fractional Hardy–Sobolev inequalities. J. Funct. Anal. 270, 4513–4539 (2016) 61. Trudinger, N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)

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Analysis and Mathematical Physics (2021) 11:117 https://doi.org/10.1007/s13324-021-00534-z

Interpolative gap bounds for nonautonomous integrals Cristiana De Filippis1 · Giuseppe Mingione2 Received: 13 December 2020 / Revised: 11 April 2021 / Accepted: 18 April 2021 / Published online: 1 June 2021 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021

Abstract For nonautonomous, nonuniformly elliptic integrals with so-called ( p, q)-growth conditions, we show a general interpolation property allowing to get basic higher integrability results for Hölder continuous minimizers under improved bounds for the gap q/ p. For this we introduce a new method, based on approximating the original, local functional, with mixed local/nonlocal functionals, and allowing for suitable estimates in fractional Sobolev spaces. Keywords Regularity · Non-autonomous functionals · (p, q)-growth Mathematics Subject Classification 35J60 · 35J70

1 Introduction In this paper we give new contributions to the regularity theory of minima of integral functionals of the type  1,1 (, R N ) Wloc

 w → F(w, ) :=



F(x, Dw) dx .

(1.1)

Here, as in the following,  ⊂ Rn will denote an open subset, where n ≥ 2 and N ≥ 1; the function F :  × R N ×n → [0, ∞) will always be Carathédory integrand. To Vladimir Gilelevich Maz’ya, master of analysis.

B

Giuseppe Mingione [email protected] Cristiana De Filippis [email protected]

1

Dipartimento di Matematica “Giuseppe Peano”, Università di Torino, Via Carlo Alberto 10, 10123 Torino, Italy

2

Dipartimento SMFI, Università di Parma, Viale delle Scienze 53/a, Campus, 43124 Parma, Italy

Chapter 4 was originally published as De Filippis, C. & Mingione, G. Analysis and Mathematical Physics (2021) 11:117. https://doi.org/10.1007/s13324-021-00534-z. Reprinted from the journal

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The definition of (local) minimizer we use here is standard in the literature and it is given by 1,1 (, R N ) is a local minimizer of the functional F in (1.1) Definition 1 A map u ∈ Wloc ˜  , we have F(u; ) ˜ < ∞ and F(u; ) ˜ ≤ F(w; ) ˜ if, for every open subset  1,1 ˜ N holds for every competitor w ∈ u + W0 (; R ).

We will abbreviate local minimizer simply by minimizer. The main point in this paper is that the integrand F : ×R N ×n → [0, ∞) is both nonautonomous and nonuniformly elliptic. Specifically, following the notation used in [20,21], we are in the situation when the ellipticity ratio R F (z, B) ≡ R(z, B) :=

supx∈B highest eigenvalue of ∂zz F(x, z) inf x∈B lowest eigenvalue of ∂zz F(x, z)

(1.2)

(B ⊂  is any ball), is such that R F (z, B) → ∞ when |z| → ∞ for at least one ball B. This happens for instance in the paramount example given by the double phase functional 

w →

 

 |Dw| p + a(x)|Dw|q dx , where 1 < p < q , 0 ≤ a(·) ∈ C α () , α ∈ (0, 1] .

(1.3)

Indeed, on a ball B such that B ∩ {a(x) = 0} = 0, in the case of (1.3) we have R(z, B) ≈ a L ∞ (B) |z|q− p + 1 .

(1.4)

The functional in (1.3) has been introduced by Zhikov [60–63] in the setting of Homogenization theory [63]. For minima of (1.3), there is a by a now a rather complete regularity theory, initiated many years ago in [25]; see [3] for the most updated statements. In particular, it has been proved that the condition on the gap q/ p α q ≤1+ , p n

(1.5)

is necessary [25,27] and sufficient [3] for the local Hölder gradient continuity of minima. Failure of (1.5) implies that in general, minimizers, that by definition are W 1, p -regular, do not belong to W 1,q . Therefore even the initial integrability bootstrap fails. The bound in (1.5) reflects the delicate balance between the smallness of a(·) around {a(x) = 0} ∩ B, and the growth of F(x, z) with respect to the gradient variable z, which is necessary to keep R(z, B) under control when proving a priori estimates for minima. We refer to [3,17,53] for a larger discussion. Functionals of the type in (1.3) fall in the realm of so-called functionals with ( p, q)growth following Marcellini’s terminology [44–46], i.e. those satisfying unbalanced ellipticity conditions of the type (for |z| large) |z| p−2 Id  ∂zz F(·, z)  |z|q−2 Id ⇒ R F (z, B)  |z|q− p 68

(1.6)

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for 1 < p ≤ q. We report in Sect. 1.3 a brief review on this kinds of problems. A distinctive feature of such functionals is that, in general, gap bounds of the type q < 1 + o(n) , p

o(n) ≈

1 n

(1.7)

are necessary and sufficient for regularity of minima [25,45]. Note that the bound appearing in (1.5) is of the type in (1.7). This said, an interesting phenomenon of interpolative nature appears when considering a priori more regular minimizers. For instance, assuming that minima are bounded, leads to non-dimensional bounds on the distance q − p, that can be made independent of n; see for instance [3,14,16]. In particular, in [17] the authors have proved that assuming that minima are bounded allows to replace (1.5) by q ≤ p+α.

(1.8)

This is better than (1.5) provided p ≤ n (that is when boundedness of minima is not automatically implied by Sobolev-Morrey embedding, and it is therefore a genuine assumption). The bound in (1.8) is optimal as shown by the counterexamples in [25,27]. A partial generalization of the result in [3] has been obtained in [20], under the same bound in (1.8), with strict inequality. More recently, in the specific case of the double phase functional (1.3), in [3] the authors have proved that, assuming a priori C 0,γ regularity for 0 < γ < 1, minimizers are regular provided q < p+

α , 1−γ

(1.9)

thus providing a further weakened gap bound. This condition is sharp too, as recently proved in [2]. It gives back (1.8) for γ → 0. In particular, note that the asymptotic of (1.9) for γ → 1 is of the type q < p + O(γ ) ,

O(γ ) ≈

1 . 1−γ

(1.10)

This is in accordance with the fact that the focal point in regularity for ( p, q)-problems is Lipschitz continuity of minima. Once this is achieved, the functional in question goes back to the realm of uniformly elliptic ones as growth conditions for large |z| become irrelevant; see also [20, Sect. 6]. Bounds of the type in (1.9) have an interpolative nature. In fact, in the scheme of Caccioppoli type inequalities coming up in regularity estimates, the a priori, assumed regularity on minima, allows for a better control of nonuniform ellipticity. The above mentioned result in [3] holds in the scalar case N = 1 and for the specific functional in (1.3); these facts play a crucial role in the analysis there. Therefore the question arises of whether or not bounds with asymptotics as in (1.10) imply regularity of C 0,γ -minima for general functionals with ( p, q)-growth. The question is open already in the autonomous case F(x, Dw) ≡ F(Dw). We are here able to give a first, positive answer by showing that bounds as in (1.10) are in general sufficient to prove Reprinted from the journal

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the basic step of improving the regularity of C 0,γ -minima from W 1, p to W 1,q . See condition (1.18) below. This initial integrability bootstrap is usually the starting point from proving higher regularity—see [20,25]—and it is essentially the best possible result in the vectorial case considered here. 1.1 Statements of the results In order to quantify ellipticity, we will use assumptions that are more general of those using the Hessian of F(·), as in (1.6), following the approach in [20,25]. In fact, rather than prescribing ellipticity of F(·) using ∂zz F(·), we will only require a quantified form of monotonicity, as in (1.11)2 . No growth assumption from above will be required of ∂zz F(·); the one on F(·) will be sufficient. Specifically, we assume that F :  × R N ×n → R is C 1 -regular in the gradient variable z and we require that ⎧ p ≤ F(x, z) ≤ L(|z|q + 1) ⎨ ν|z|  2 ( p−2)/2 ν |z 1 | + |z 2 |2 + μ2 |z 1 − z 2 |2 ≤ (∂z F(x, z 1 ) − ∂z F(x, z 2 )) · (z 1 − z 2 ) ⎩ |∂z F(x, z) − ∂z F(y, z)| ≤ L|x − y|α (|z|q−1 + 1) (1.11) hold whenever x, y ∈ , z, z 1 , z 2 ∈ R N ×n , where 1 < p ≤ q, μ ∈ [0, 1], α ∈ (0, 1] and 0 < ν ≤ 1 ≤ L are fixed constants. Note that (1.11)1 implies that minimizers of F are automatically locally W 1, p -regular. As we are dealing with a nonautonomous functional, the so-called Lavrentiev phenomenon naturally comes into the play [43,60– 62]. Its nonoccurrence is a necessary condition for regularity of minima. Therefore we are led to consider a functional, called Lavrentiev gap, providing a quantitative measure of such a phenomenon. We refer the reader to [1,20,25] for more information. Since we are in fact interested in the regularity of a priori Hölder continuous minima of the functional (1.1), we will adopt suitable definitions of relaxed and gap functionals aiming at exploiting this fact. For this, let us consider a functional of the type in (1.1), where the integrand F(·) satisfies (1.11), and let us fix numbers numbers H > 0 and γ ∈ (0, 1). Given a ball B ⊂ , the natural relaxation of F we consider here is F¯ H ,γ (w, B) :=

inf

{w j }∈C H ,γ (w,B)

  lim inf F(x, Dw j ) dx , j→∞

(1.12)

B

defined for any w ∈ W 1,1 (B, Rn ), where (see (2.1) below for the notation) C H ,γ (w, B) := {w j } ⊂ W 1,∞ (B, R N ) : w j w in W 1, p (B, R N ),

sup [w j ]0,γ ;B ≤ H .

(1.13)

j

Accordingly, as in [1,20,25], we consider the Lavrentiev gap functional L F,H ,γ (w, B) := F¯ H ,γ (w, B) − F(w, B) , 70

(1.14)

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defined for every w ∈ W 1,1 (B) such that F(w, B) is finite; we set L F,H ,γ (w, B) = 0 otherwise. Remark 1 We here collect a few immediate consequences of the above definitions. • The convexity of F(·), implied by (1.11)2 , provides that the functional F in (1.1) is lower semicontinuous with respect to the weak convergence of W 1, p . It follows that L F,H ,γ (·, B) ≥ 0 holds whenever w ∈ W 1,1 (B, R N ) and B   is a ball. 0,γ 1, p • Consider a map w ∈ Wloc (, R N ) ∩ Cloc (, R N ). A simple mollification argument then shows that C H ,γ (w, B) is non empty whenever B   for H ≈ wC 0,γ (B) . For this see also the proofs in Sect. 3.11 below. This anticipates that, when considering a C 0,γ -regular minimizer u of the functional F, it will happen that we will use L F,H ,γ (u, B) with H ≈ uC 0,γ (B) (see for instance Theorem 2 below, and compare (1.22) with (1.19)). • A straightforward consequence of (1.12)–(1.14) is Proposition 1 Let w ∈ W 1,1 (B, R N ) be such that F(w, B) is finite, where B ⊂  is a fixed ball. Then F(w, B) = F¯ H ,γ (w, B) for some H > 0, if and only if there exists {w j } ∈ C H ,γ (w, B) such that F(w j , B) → F(w, B) .

(1.15)

• Accordingly, if C H ,γ (w, B) is empty, then F¯ H ,γ (w, B) = ∞; moreover, again in this case, if F(w, B) is finite, then F¯ H ,γ (w, B) = L F,H ,γ (w, B) = ∞. This is in accordance with the fact that when w is not C 0,γ -regular, it is in general impossible to build a sequence from C H ,γ (w, B), for any H > 0, approximating w in energy in the sense of (1.15) below. Such approximations are in fact usually built via smooth convolutions when w is already Hölder continuous. • Conversely, if C H ,γ (w, B) is non empty, then w ∈ C 0,γ (B) and [w]0,γ ;B ≤ H . • If w is a locally W 1,q ∩C 0,γ -regular map, then a density and convolution argument gives that L F,H ,γ (w, B) = 0 holds for every ball B  , with H ≈ wC 0,γ (B) + 1. This last condition is therefore in a sense necessary to prove the local W 1,q regularity of C 0,γ -regular minima of the original functional F. Accordingly to the last bullet in Remark 1, the main result of this paper is now 1, p

Theorem 1 Let u ∈ Wloc (, R N ) be a minimizer of functional (1.1), under assumptions (1.11) and min{α, 2γ } q < p+ , ϑ(1 − γ )

 where ϑ :=

1 if p ≥ 2 2 p if 1 < p < 2 ,

0 < γ < 1. (1.16)

Assume also that L F,H ,γ (u, Br ) = 0 Reprinted from the journal

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(1.17)

C. De Filippis, G. Mingione

holds for a ball Br   with r ≤ 1, and for some H > 0. If q is a number such that min{α, 2γ } ϑ(1 − γ )

q ≤q< p+

(1.18)

and B  Br is a ball concentric to Br , then Du L q (B ) ≤

 κ2 c 1/ p [F(u, B )] + H + 1 r (r − )κ1

(1.19)

holds for constants c ≡ c(n, p, q, ν, L, α, γ , q) and κ1 , κ2 ≡ κ1 , κ2 (n, p, q, α, γ , q). 1,q In particular, if (1.17) holds for every ball Br  , then u ∈ Wloc (, R N ). Remark 2 The reader might of course wonder where the a priori C 0,γ -regularity of the minimizer u is assumed in Theorem 1. This is hidden in assumption 1.17. In fact, as F(·, Du) ∈ L 1loc () by minimality, it follows from the fourth and the fifth bullet of Remark 1 that C H ,γ (u, Br ) is non-empty and therefore u ∈ C 0,γ (Br , R N ) (with [u]0,γ ;Br ≤ H ). This said, the bound in (1.18) is exactly of the type in (1.10). Let us now consider the case p ≥ 2. When α ≤ 2γ the bound in (1.16) coincides with (1.9), that is the one considered in [3] for the specific functional (1.3). As mentioned above, assumption (1.17) is in a sense necessary to prove local W 1,q regularity of minimizers of F. As a matter of fact, condition (1.17) is always satisfied in a large number of situations. A very relevant one is when the integrand is autonomous, i.e., F(x, Du) ≡ F(Du). In such a case, the Lavrentiev gap disappears due to basic convexity arguments, and we have: 1, p

0,γ

Theorem 2 Let u ∈ Wloc (, R N ) ∩ Cloc (, R N ) be a minimizer of functional (1.1), where 0 < γ < 1, under assumptions (1.11) with F(x, z) ≡ F(z) and q < p+

min{1, 2γ } , ϑ(1 − γ )

(1.20)

where ϑ is as in (1.16). If q is a number such that q ≤q< p+

min{1, 2γ } ϑ(1 − γ )

(1.21)

and B  Br   are concentric balls, then Du L q (B ) ≤

  κ2 c [F(u, Br )]1/ p + [u]0,γ ;Br + 1 κ (r − ) 1

(1.22)

1,q

holds with c, κ1 , κ2 as in (1.19). In particular, u ∈ Wloc (, R N ). Another situation when (1.17) can be automatically satisfied, is when the integrand F(·) is equivalent to a convex function G : R N ×n → [0, ∞) modulo a multiplicative 72

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factor, i.e., b(x)G(z)  F(x, z)  b(x)G(z) + 1 ,

1, p

0 ≤ b(·), 1/b(·) ∈ L ∞ () . (1.23)

0,γ

Corollary 1 Let u ∈ Wloc (, R N ) ∩ Cloc (, R N ) be a minimizer of functional (1.1) under the assumptions (1.11) and (1.16). Furthermore, assume that (1.23) is satisfied too. Then (1.22) holds whenever B  Br   are concentric balls and for the range of exponents in (1.18). Back to the full nonautonomous case, a relevant example in this setting is given by [3, Theorem 4]. This deals with functionals modelled on the double phase functional in (1.3), i.e., growth conditions as |z| p + a(x)|z|q  F(x, z)  |z| p + a(x)|z|q + 1 ,

0 ≤ a(·) ∈ C 0,α () , (1.24)

are assumed for every (x, z) ∈  × R N ×n , no matter (1.11) are satisfied or not. Then (1.9) guarantees that the approximation in energy (1.15) holds for a sequence of W 1,∞ -regular maps {w j }, provided w ∈ C 0,γ holds and the bound in (1.9) is in force. This fact allows to draw another consequence from Theorem 1, that is 1, p

0,γ

Corollary 2 Let u ∈ Wloc (, R N ) ∩ Cloc (, R N ) be a minimizer of functional (1.1) under the assumptions (1.11) and (1.16). Furthermore, assume that (1.24) is satisfied too. Then (1.22) holds for the range of exponents displayed in (1.18). More general cases of double sided bounds as in (1.24) for which similar corollaries hold can be found in [25, Sect. 5]. We refer to this last paper also a for larger discussion on the use of Lavrentiev gap functionals in this setting. 1.2 Novelties and techniques The technique leading to the proof of Theorem 1 makes use of three main ingredients. The first one is a method aimed at approximating, on a fixed ball Br  , the original minimizer u of F with a sequence {u j } of W 1,q -regular solutions to a different kind of variational problems. This is necessary, as the starting lack of W 1,q -integrability of u does not allow to use the Euler–Lagrange system of F. The possibility of this approximation relies on assumption (1.17), that can be used, in a sense, to find good boundary values to build the approximating minimizers {u j }. Here we encounter a first difficulty as, in order to get uniform a priori integrablity estimates on {Du j }, we would need that the sequence {u j } is bounded in C 0,γ , exactly as u (see Remark 2). We will actually build the sequence {u j } in a way that it is bounded in a fractional Sobolev-Slobodevsky space W s,2d . For our purposes this is actually sufficient, as these spaces are in a sense a good approximation of C 0,γ , once proper choices of s close to γ and d large enough, are made. To find {u j }, we employ a novel approximation using additional nonlocal terms, i.e., adding a suitable truncated Gagliardo-type seminorm Reprinted from the journal

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term to the original functional F, together with a more standard L 2d -penalization term. Specifically, we consider minimizers {u j } of perturbed functionals of the type 



w → F(w, Br ) + ε  +

dx +

Br

Br

(|w|2 − M02 )d+ dx

 d |w(x) − w(y)|2 − M 2 |x − y|2γ +

 Rn

|Dw|

2d

|x − y|n+2sd

Rn

dx dy

(1.25)

for ε ≡ ε j small, s close to γ and suitably large d, M0 and M. The last two constants depend on uC 0,γ , which is finite by assumption. Here, following a standard notation, we are denoting (t − k)+ := max{t − k, 0} ,

fort, k ∈ R .

(1.26)

Functionals of mixed local/nonlocal type, in the quadratic/linear case F(x, z) ≡ |z|2 , have been recently studied in [6,7] under special boundary conditions. As fas as we know, this is the first paper where nonlinear functionals of the type in (1.25) are considered, and a priori estimates are presented. Let us mention that in the setting of functionals with ( p, q)-growth conditions, purely local approximations aimed at using the L ∞ -information, have been considered for the first time in [14] in the autonomous case; see also [20] for the nonautonomous case under assumptions (1.11). These approximations are made using only the first line in (1.25), and are not suitable to preserve the initial C 0,γ -regularity information, or at least some part of it. Adding the last line in (1.25) therefore turns out to be crucial. On the other hand this turns out to generate different a priori estimates. The second ingredient, which is taken from [20,25], is a suitable use of the difference quotients techniques in the setting of Fractional Sobolev spaces. At this stage, the Hölder continuity of ∂z F(·) in (1.11)3 is automatically read as a fractional differentiability and allows to get uniform estimates in Nikolski spaces for Du j . We finally use the last and third ingredient. The assumed Hölder continuity of u implies that {u j } is uniformly bounded in W s,2d , as mentioned above. This allows to use a Gagliardo-Nirenberg type interpolation inequality on Du j , which improves the exponents intervening when using fractional Sobolev embedding theorem in the setting of Caccioppoli type inequalities; see Lemma 3 below. Ultimately, we get uniform higher integrability estimates on the sequence {Du j }, that finally imply the higher integrability of Du in (1.19). 1.3 Brief review on nonuniformly elliptic functionals Functionals with ( p, q)-growth fall in the larger realm of nonuniformly elliptic problems. In the case of autonomous integrals of the type  F(Dw) dx ,

w→

(1.27)

B

74

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the nonuniform ellipticity of F(·) amounts to assert that the ellipticity ratio sup R F (z) := z

highest eigenvalue of ∂zz F(z) lowest eigenvalue of ∂zz F(z)

(1.28)

becomes unbounded when |z| → ∞. This is in accordance to the analogous, nonautonomous definition in (1.2), that in fact reduces to (1.28) in the autonomous case. We refer to [36,40,57] for this definition. As for equations, nonuniform ellipticity originally stems from some remarkable models, as for instance the minimal surface equation. It is today a classical topic of its own in pde theory. Early work in this line include those of Ladyzhenskaya and Uraltseva [40,41], Hartman and Stampacchia [29], Trudinger [57,58], Ivanov [34,35], Ivoˇckina and A.P. Oskolkov [37]. In particular, Serrin’s landmark paper [56] has established important solvability and regularity results, and Ivanov’s monograph [36] contains a thoroughly exposition of the theory until the beginning of the eighties. These works deal with elliptic equations, also in nondivergence form, and feature a number of a priori estimates and existence theorems under various structural conditions. The variational theory is probably the best setting where to frame nonuniform ellipticity, without knowing additional strucures. In fact, when considering elliptic equations, there is always a problem in identifying the correct set of test functions one can use. The well-known difference between distributional and energy solutions becomes wider. In particular, when considering an elliptic equation of the type div a(Du) = 0, it is crucial to have the possibility to test the distributional form with functions that are proportional to the solution u. In the case of ( p, q)-growth conditions, this is possible only when u ∈ W 1,q , but this is not verified in general. This leads, when dealing with equations, to establish, simultaneously, existence and regularity theorems for solutions, as for instance shown in [4,45]. This ambiguity, widely discussed by Zhikov [60,61], is not present in the case of minimizers, where minimality allows to work out approximation procedures. These are of the type also considered in this paper. These approximations schemes are able to select all minimizers in the autonomous case (and all minimizers of the relaxed functional in nonautonomous one). Once combined with suitable a priori estimates, such approximation schemes lead to regularity results for minimizers. After a first pioneering paper of Uraltseva and Urdaletova [59], functionals with ( p, q)-growth have been systematically investigated by Marcellini [44–46], whose work started what is a by now rather vast literature on the subject. For this, we refer to the surveys [48,49,52,53] for a reasonable overview. Typical examples of functionals with ( p, q)-growth are   w →

Reprinted from the journal

|Dw| + p



n 

 ai (x)|Di w|

i=1

75

pi

dx ,

C. De Filippis, G. Mingione

where 1 ≤ ai (x) ≤ L and 1 < p = p1 ≤ p2 ≤ · · · ≤ pn = q,  w → w → w →











|Dw| p(x) dx , 1 < p ≤ p(x) ≤ q , |Dw| p(x)B(|Dw|) dx, 1 < p ≤ p(x) ≤ q − γ , 1 ≤ B(|z|) ≤ γ , |Dw| p(x) B(|Dw|) dx ,

where 1 < p ≤ p(x) ≤ q − q1 and 1 ≤ B(|z|) ≤ L(1 + |z|q1 ). Functionals of the type in the above displays typically come up in applied settings, where standard growth conditions are not sufficient to catch important aspects in modelling. We refer for instance to applications in the theory of Homogenization [60–63], Image segmentation [31], Elasticity [43] and non-Newtonian Fluid mechanics. For more on applications we refer to [52,53]. Moreover, connections emerge with abstract theory of functions spaces and Harmonic Analysis. It that case it happens that assumptions implying regularity of minimizers naturally connect with those implying good functional theoretic properties such as density of smooth functions and boundedness of maximal and integral operators. See [32] and [53, Sect. 6]. As emphasized above, the gap q/ p is a crucial quantity to measure the rate of nonuniform ellipticity, that is the growth of R F (z) in (1.28). In this case we have growth conditions as in (1.4) and therefore we have polynomial nonuniform ellipticity. Gap bounds of the type in (1.7) are always essential, also in the parabolic case [9]. It is important to remark that such gap bound conditions play a basic role both in the scalar case N = 1, and in the vectorial one N > 1. In the latter, partial regularity comes into the play. This means that minimizers are proved to be regular only outside a closed, negligible subset, with (sometimes) the possibility to estimate the Hausdorff dimension of the singular set. Results of this type are available both in the convex case [15] and in the quasiconvex one [55]; see also the recent [54]. We refer to these last three papers for a list of references and again to [52] for an overview on partial regularity and singular set dimension reduction results. We note that a great deal of work has been devoted to find better bounds on q/ p, amongst those of the type (1.7), starting with [44,45]. Bounds under nonautonomous structures have been found and/or used in [8,13,18,19,30–32,38]. Recent, interesting work using adapted cut-off functions has been realised in [5,33,54], allowing to get sharp bounds in the case one is interested in finding conditions implying boundedness of u rather than that of Du. Better bounds for gradient L ∞ -estimates can be obtained as well. Improved bounds also occur under special structures [10]. Polynomial nonuniform ellipticity is not the only type of nonuniform ellipticity. Functionals with faster growth emerged over the years several times in the literature. A model prototype is provided by  w →



exp(|Dw| p ) dx ,

76

p≥1.

(1.29)

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Interpolative gap bounds for nonautonomous integrals

Exponential growth functionals as in (1.29) are classical in the Calculus of Variations starting by the work of Duc and Eells [24], Lieberman [42], and Marcellini [46]. They are treated for instance in the setting of weak KAM-theory in [26]. Even faster growth conditions can be considered by allowing arbitrary compositions of exponentials, i.e.,  w →



exp(exp(. . . exp(c(x)|Dw| p ) . . .)) dx ,

p ≥ 1 , 0 < ν ≤ c(·) ≤ L . (1.30)

When considering the functional in (1.30) in the autonomous case c(·) ≡ 1, we have that [4, (6.13)] R F (z)  t p−1 exp(exp(· · · exp(|z| p ) · · · )) + 1

(1.31)

where, if k ≥ 1 is the number of composing exponentials involved in (1.30), the number in (1.31) is k − 1 (it is zero in the case of (1.29)). Regularity results for functionals as in (1.29) can be found in [4,46,47], to which we refer for further references. For the nonautonomous case, i.e., when c(·) in (1.30) is not a constant, more delicate conditions emerge. A typical result in this setting claims that minimizers of (1.30) are locally Lipschitz provided c(·) ∈ W 1,d , for some d > n. For this kind of theorems, we refer to [21].

2 Preliminaries 2.1 Notation In the rest of the paper, we denote by  ⊂ Rn an open subset, n ≥ 2. We denote by c a general constant larger than one. Different occurrences from line to line will be still denoted by c. Special occurrences will be denoted by c1 , c2 , c˜ or likewise. When a relevant dependence on parameters occurs, this will be emphasized by putting the correspondent parameters in parentheses. Finally, the symbol  denotes inequalities where absolute constants are involved. As usual, we denote by Br (x0 ) := {x ∈ Rn : |x − x0 | < r }, the open ball with center x0 and radius r > 0; when it is clear from the context, we omit denoting the center, i.e., Br ≡ Br (x0 ). When not otherwise stated, different balls in the same context will share the same center. We will also denote B1 = B1 (0) if not differently specified. Finally, with B being a given ball with radius r and σ being a positive number, we denote by σ B the concentric ball with radius σ r . In denoting several function spaces like L p (), W 1, p (), we will denote the vector valued version by L p (, Rk ), W 1, p (, Rk ) in the case the maps considered take values in Rk , k ∈ N. Sometimes we will abbreviate L p (, Rk ) ≡ L p (), W 1, p (, Rk ) ≡ W 1, p (). With B ⊂ Rn being a measurable subset with bounded positive measure 0 < |B| < ∞, and with w : B → Rk , being a measurable Reprinted from the journal

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C. De Filippis, G. Mingione

map, we will denote the integral average of w over B by   1 (w)B ≡ − w(x) dx := w(x) dx . |B| B B Given z, ξ ∈ R N ×n , their Frobenius product is defined as z · ξ = z iα ξiα ; it follows that ξ · ξ = |ξ |2 and in the rest of the paper we will use the classical Frobenius norm for matrixes. We will use a similar notation for the scalar product in R N . As usual, the symbol ⊗ denotes the tensor product; in particular, given λ ∈ Rn and ι ∈ Rn , we have ι ⊗ λ ≡ {ια λi } ∈ R N ×n , 1 ≤ i ≤ n, 1 ≤ α ≤ N . In this paper we use the standard notation [w]0,γ ;B :=

sup

x,y∈B,x = y

|w(x) − w(y)| , |x − y|γ

(2.1)

whenever B ⊂ Rn is a subset, γ ∈ (0, 1] and w : B → Rk . Accordingly, the C 0,γ norm of w is defined by wC 0,γ (B) := w L ∞ (A) + [w]0,γ ;B . 2.2 Fractional spaces and interpolation inequalities We collect here some basic facts about fractional Sobolev spaces. We refer to [23, 25] for more results. For a map w :  → Rk and a vector h ∈ Rn , we denote by τh : L 1 (, Rk ) → L 1 (|h| , Rk ) the standard finite difference operator pointwise defined as (τh w)(x) ≡ τh w(x) := w(x + h) − w(x) ,

(2.2)

whenever |h| := {x ∈  : dist(x, ∂) > |h|} is not empty. We will consider a similar operator acting on maps ψ :  ×  → Rk , this time defined by (τ˜h w)(x) ≡ τ˜h ψ(x, y) := ψ(x + h, y + h) − ψ(x, y) .

(2.3)

Definition 2 Let α0 ∈ (0, ∞) \ N, p ∈ [1, ∞), k ∈ N, n ≥ 2, and let  ⊂ Rn be an open subset. • If α0 ∈ (0, 1), the fractional Sobolev space W α0 , p (, Rk ) consists of those maps w :  → Rk such that the following Gagliardo type norm is finite:  

|w(x) − w(y)| p dx dy n+α0 p   |x − y| =: w L p () + [w]α0 , p; .

1/ p

wW α0 , p () := w L p () +

(2.4)

In the case α0 = [α0 ] + {α0 } ∈ N + (0, 1) > 1, we have w ∈ W α0 , p (, Rk ) iff wW α0 , p () := wW [α0 ], p () + [D [α0 ] w]{α0 }, p; 78

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Interpolative gap bounds for nonautonomous integrals α ,p

is finite. The local variant Wloc0 (, Rk ) is defined by requiring that w ∈ α ,p ˜ Rk ) for every open subset  ˜  . Wloc0 (, Rk ) iff w ∈ W α0 , p (, • For α0 ∈ (0, 1], the Nikol’skii space N α0 , p (, Rk ) is defined by prescribing that w ∈ N α0 , p (, Rk ) if and only if  w N α0 , p (,Rk ) := w L p (,Rk ) +

 sup

|h| =0

|h|

|w(x + h) − w(x)| p dx |h|α0 p

α ,p

1/ p .

α ,p

The local variant Nloc0 (, Rk ) is defined analogously to Wloc0 (, Rk ). We have that W α0 , p (, Rk )  N α0 , p (, Rk )  W β, p (, Rk ), for every β < α0 , hold for sufficiently domains . These inclusions are somehow quantified in the following lemma Lemma 1 Let w ∈ L p (), p ≥ 1, and assume that for α0 ∈ (0, 1], S ≥ 0 and an ˜   we have that τh w p ˜ ≤ S|h|α0 holds for every open and bounded set  L () ˜ ∂). Then w ∈ W β, p (, ˜ Rk ) h ∈ Rn satisfying 0 < |h| ≤ d, where 0 < d ≤ dist(, for every β ∈ (0, α0 ), and the estimate 

wW β, p () ˜

w L p () d (α0 −β) S ˜ ≤c + 1/ p (α0 − β) min{d n/ p+β , 1}

 (2.5)

holds with c ≡ c(n, p). In particular, let B  Br ⊂ Rn be concentric balls with r ≤ 1, w ∈ L p (Br , Rk ), p > 1 and assume that, for α0 ∈ (0, 1], S ≥ 1, there holds τh w L p (B ,Rk ) ≤ S|h|α0

for every h ∈ Rn with 0 < |h| ≤

r − , where K ≥ 1 . K (2.6)

Then it holds that wW β, p (B ,Rk ) ≤

c (α0 − β)1/ p



r − K

α0 −β

 S+c

K r −

n/ p+β w L p (Br ,Rk ) , (2.7)

where c ≡ c(n, p). Proof A main point in Lemma 1, is the precise quantitative linkage between the size of |h| appearing in (2.6), and the dependence on the constants appearing in (2.7). This will be crucial in the applications we will make of it. See Sects. 3.9 and 3.10 below. For this reason we decide to report the easy proof, since it does not appear in the literature Reprinted from the journal

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C. De Filippis, G. Mingione

but is often reported as folklore. Fubini’s theorem yields  

  |w(x) − w(y)| p |w(x + h) − w(x)| p dx dy ≤ dx dh n+β p |x − y| |h|n+β p ˜ ∩{|x−y| 1 and θ ∈ (0, 1) be such that θ 1−θ 1 = + . p˜ a t

1 = θ s1 + (1 − θ )s2 ,

Then every function w ∈ W s1 ,a (Br )∩W s2 ,t (Br ) belongs to W 1, p˜ (B ) and the inequality Dw L p˜ (B ) ≤

c [w]θs1 ,a;Br Dw1−θ W s2 −1,t (Br ) (r − )κ

(2.9)

holds for constants c, κ ≡ c, κ(n, s1 , s2 , a, t). Next, a classical iteration lemma of [28, Lemma 6.1], that is Lemma 4 Let Z : [ , r ] → R be a nonnegative and bounded function, and let θ ∈ (0, 1) and A ≥ 0, γ1 ≥ 0 be numbers. Assume that Z(t) ≤ θ Z(s) +

A (s − t)γ

holds for ≤ t < s ≤ r . Then the following inequality holds with c ≡ c(θ, γ1 ): Z( 0 ) ≤

cA , (r − )γ

3 Proof of Theorems 1, 2 and Corollaries 1, 2 These proofs take twelve different steps. The first ten are dedicated to the proof of Theorem 1. In Step 11 we deal with Corollaries 1, 2 and Step 12 is dedicated to Theorem 2. 3.1 Step 1: Choice of parameters Fix q as in (1.18). We start considering parameters s, d and β initially satisfying 0≤s max{q, n} ,

0 < β < min{α, 2γ } .

(3.1)

Accordingly, we define the function p˜ ≡ p(s, ˜ d, β) as p˜ : =

2d [ p(1−s)+β] if p ≥ 2 β+2d(1−s)

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p˜ :=

and

81

2dp[2(1 − s) + β] if 1 < p < 2 . pβ + 4d(1−s) (3.2)

C. De Filippis, G. Mingione

Note that the inequality d > p/2, which is true by (3.1), in particular implies p˜ < 2d .

(3.3)

The function p˜ is increasing in all its variables (we again use that d ≥ p/2 for this), with lim

s→γ ,d→∞,β→min{α,2γ }

p(s, ˜ d, β) = p +

min{α, 2γ } , ϑ(1 − γ )

(3.4)

where ϑ has been defined in (1.16). In the following, we always take d such that d>

n max{q, n} ⇒ β0 := s − > 0. 2s 2d

(3.5)

In view of (3.4) we further increase both s and d, while still keeping (3.5), and find β such that β < α0 := min{α, 2β0 } < min{α, 2γ }

(3.6)

and q ≤ q < p(s, ˜ d, β) < p +

min{α, 2γ } . ϑ(1 − γ )

(3.7)

Note that, by further increasing s, d, β still in the range fixed in (3.1), conditions (3.5)–(3.7) still hold (as noted before, p(s, ˜ d, β) is increasing with respect to all its variables). Keeping this in mind, we next distinguish two cases. When p ≥ 2, note that 1−s p(s, ˜ d, β)(q − p) p(s, ˜ d, β) − p p(1 − s) + β (q − p)(1 − γ ) (1.16) 2d(q − p)(1 − s) = = lim < 1. s→γ ,d→∞,β→min{α,2γ } β(2d − p) min{α, 2γ } lim

s→γ ,d→∞,β→min{α,2γ }

Therefore, we finally again increase s, d and β again, in order to have 1−s p(s, ˜ d, β)(q − p) < 1. p(s, ˜ d, β) − p p(1 − s) + β

(3.8)

When instead 1 < p < 2, we note that 2(1 − s) p(s, ˜ d, β)(q − p) p(s, ˜ d, β) − p p[2(1 − s) + β] 2(q − p)(1 − γ ) (1.16) 4d(q − p)(1 − s) lim = < 1, s→γ ,d→∞,β→min{α,2γ } pβ(2d − p) p min{α, 2γ } lim

s→γ ,d→∞,β→min{α,2γ }

82

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and therefore we again find s, d and β such that p(s, ˜ d, β)(q − p) 2(1 − s) < 1. p(s, ˜ d, β) − p p[2(1 − s) + β]

(3.9)

From now on we will always consider this final choice of s, d and β, and therefore we will use (3.2)–(3.9) for the rest of the proof. In view of this, from now on we will express any dependency from (s, d, β) as a dependence on ( p, q, α, γ , q). In the following, in order to shorten the notation, we will denote data ≡ (n, N , p, q, ν, L, α, γ , q) ,

datae ≡ (n, p, q, α, γ , q) .

(3.10)

3.2 Step 2: Hölder and Sobolev extensions Let us note that in the definition 1.12 we can replace C H ,γ (w, B) in (1.13) by 

{w j } ⊂ W

1,∞

(B, R ) : (w j ) B =0, w j w in W N

1, p

(B, R ), sup [w j ]0,γ ;B ≤H . N

j

Such a replacement leaves the values of F¯ H ,γ and L F,H ,γ unaltered. Keeping this fact in mind, we fix a ball Br ⊂ B2r   with r ≤ 1 and such that (1.17) holds, and, by Proposition 1, we can find a sequence {u˜ j } ⊂ W 1,∞ (Br , R N ) ∩ C 0,γ (Br , R N ),

(u˜ j ) Br = 0

(3.11)

such that, eventually passing to a not relabelled subsequence, it holds that ⎧ u˜ j u weakly in W 1, p (Br , R N ), u˜ j → u strongly in L p (Br , R N ) ⎪ ⎪ ⎪ ⎨F(u˜ , B ) → F(u, B ) j r r −γ u˜  ∞ −γ −n/ p u˜  p ⎪ + [ u ˜ r j L (Br ) j ]0,γ ;Br + r j L (Br ) ≤ cH ⎪ ⎪ ⎩ p νD u˜ j  L p (Br ) ≤ F(u˜ j , Br ) ≤ F(u, Br ) + 1

(3.12)

with the last two lines that hold for every j ≥ 1, and where c ≡ c(n, p) is an absolute constant. All the facts from (3.11) are directly coming from Proposition 1 but (3.12)3 , that maybe deserves a few words; there only [u˜ j ]0,γ ;Br ≤ H directly comes from the definition of the Lavrentiev gap. For this, note that, if x ∈ Br , then by (3.11) we find, by Jensen’s inequality, that |u˜ j (x)| = |u˜ j (x) − (u˜ j ) Br | ≤ cr γ [u˜ j ]0,γ ;Br ≤ cr γ H . This implies (3.12)3 . Now, by [51, Theorem 2], we can extend the u˜ j ’s to the whole Rn by determining maps {u¯ j } such that [u¯ j ]0,γ ;Rn ≤ c[u˜ j ]0,γ ;Br ≤ cH

(3.13)

for an absolute constant c which is independent of j. Let η¯ ∈ C01 (B3r /2 ) be such that 1 B¯ r ≤ η¯ ≤ 1 B¯ 3r /2 and |D η| ¯  1/r Reprinted from the journal

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(3.14)

C. De Filippis, G. Mingione

and set v¯ j := u¯ j η. By this very definition, (3.13) and (3.14) we have that ⎧ r −γ v¯ j  L ∞ (Rn ) ≤ c∗ H ⎪ ⎪ ⎪ ⎪ ⎪ [v¯ j ]0,γ ;Rn ≤ c∗ H ⎪ ⎪ ⎪ ⎨ v¯ ≡ u¯ ≡ u˜ in B j j j r ⎪ supp v ¯  B j 3r /2 ⎪ ⎪ ⎪ ⎪ ⎪ v¯ j ∈ W 1,2d (Br , R N ) ⎪ ⎪ ⎩ s r [v¯ j ]s,2d;Rn ≤ c∗r γ +n/(2d) H

(3.15)

hold for every j ∈ N, an absolute constant c∗ ≡ c∗ (n, d, γ , s) ≡ c∗ (datae ), which is independent of j. We confine ourselves to sketch the simple proofs of (3.15)1,2 and of (3.15)6 , the other assertions being a direct consequence of the definition of v¯ j (recall also (3.11)). Let us first remark that, triangle inequality, (3.13) and finally (3.12)3 , imply u¯ j  L ∞ (B2r ) ≤ c[u˜ j ]0,γ ;Br r γ + u˜ j  L ∞ (Br ) ≤ cHr γ ,

(3.16)

that is (3.15)1 . In order to prove (3.15)2 , it is sufficient to note that if x, y ∈ B3r /2 , then, using (3.13) and (3.16) it follows |v¯ j (y) − v¯ j (x)| = |u¯ j (y)η(y) − u¯ j (x)η(x)| ≤ |u¯ j (y) − u¯ j (x)| + |η(y) − η(x)|u¯ j  L ∞ (B2r )

≤ c[u¯ j ]0,γ ;B2r |x − y|γ + Dη L ∞ (Br ) u¯ j  L ∞ (B2r ) |x − y| c ≤ cH |x − y|γ + γ u¯ j  L ∞ (B2r ) |x − y|γ r ≤ c∗ H |x − y|γ .

As it is v j ≡ 0 outside B3r /2 , this is sufficient to conclude with (3.15)2 . Finally, the proof of (3.15)6 . By (3.15)1,2,3,4 , we have  [v¯ j ]2d s,2d;Rn = 2

 Rn \B2r

B2r



 +

B2r

 =2

B2r



Rn \B2r





|v¯ j (x) − v¯ j (y)|2d dx dy |x − y|n+2sd

|v¯ j (x) − v¯ j (y)|2d dx dy |x − y|n+2sd B3r /2

|v¯ j (x)|2d dx dy |x − y|n+2sd

|v¯ j (x) − v¯ j (y)|2d dx dy |x − y|n+2sd B2r B2r dz v¯ j 2d L 2d (B3r /2 ) n+2sd Rn \Br /2 |z|   dx dy +c[v¯ j ]2d 0,γ ;B2r n+2d(s−γ ) B2r B2r |x − y|

+  ≤c

84

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Interpolative gap bounds for nonautonomous integrals

≤

cr n+2d(γ −s) H 2d ≡ cr n+2d(γ −s) H 2d s(γ − s)

(3.17)

for c ≡ c(datae ), and the proof of (3.15)6 follows. 3.3 Step 3: Approximation via nonlocal functionals We introduce the nonlocal Dirichlet class    X(v¯ j , Br ) := v ∈ v¯ j + W01,2d (Br , R N ) ∩ W s,2d (Rn , R N ) : v ≡ v¯ j on Rn \ Br . This is a convex, closed subset of W01,2d (Br , R N ) ∩ W s,2d (Rn , R N ), and it is nonempty, as v¯ j ∈ X(v¯ j , Br ) by (3.15). Next, we define u j ∈ X(v¯ j , Br ) as the solution to u j →

min

w∈X(v¯ j ,Br )

F j (w, Br ) ,

(3.18)

where, keeping in mind the notation in (1.26), it is  F j (w, Br ) := F(w, Br ) + ε j (|Dw|2 + μ2 )d dx Br  (|w|2 − M02 )d+ dx +  +

Br

Rn

 d |w(x) − w(y)|2 − M 2 |x − y|2γ +



|x − y|n+2sd

Rn

dx dy (3.19)

with ⎧ 1 ⎪ ⎪  ⎨ ε j :=  4d D v¯ j  L 2d (B ) + j + 1 r ⎪ ⎪ ⎩ M := 16c r γ H , M := 16c H . 0

∗

(3.20)

∗

In (3.20) c∗ ≡ c∗ (datae ) is the same (absolute) constant has been defined in (3.15). Let us briefly point out how Direct Methods of the Calculus of Variations apply here to get the existence of u j in (3.18); the main point is essentially to prove that the functional is coercive on X(v¯ j , Br ). With j ∈ N being fixed, let us consider a minimizing sequence {w j,k }k ⊂ X(v¯ j , Br ), i.e., such that lim F j (w j,k , Br ) = k

Reprinted from the journal

inf

w∈X(v¯ j ,Br )

85

F j (w, Br ) .

(3.21)

C. De Filippis, G. Mingione

We now have (recall that w j,k ≡ v¯ j outside Br , v¯ j ≡ 0 outside B3r /2 and v¯ j ≡ u˜ j in Br ) w j,k 2d ≤ v¯ j 2d L 2d (Rn ) L 2d (B

3r /2 )

+ w j,k 2d L 2d (B

r)

≤ cr n+2dγ H 2d + w j,k − v¯ j 2d L 2d (B ≤ cr

n+2dγ

cr 2d F

≤

H

2d

+ cr

j (w j,k ,

2d

Br )

εj

r) 2d Dw j,k  L 2d (B ) r

+ cr 2d D v¯ j 2d L 2d (B

r)

+ cr 2d D u˜ j 2d + cr n+2dγ H 2d L 2d (B ) r

(3.22)

for c ≡ c(datae ); in the second estimate in the above display we have used also (3.15)1 and in the last line the very definition of F j from (3.19). Furthermore, triangle inequality implies 

[w j,k ]2d s,2d;B2r

≤c



(|w j,k (x) − w j,k (y)|2 − M 2 |x − y|2γ )d+ dx dy |x − y|n+2sd B2r B2r   dx dy + cM 2d n−2d(γ −s) |x − y| B2r B2r

≤ cF j (w j,k , Br ) +

cM 2d r n+2d(γ −s) γ −s

= cF j (w j,k , Br ) + cr n+2d(γ −s) H 2d ,

(3.23)

where c ≡ c(datae ) and we have used the definition of M from (3.20)2 in the last line. In turn, using (3.15)1 , and (3.22)-(3.23), we have  [w j,k ]2d s,2d;Rn

=2



Rn \B2r

 =2



Rn \B2r



+2

Rn \B2r

B2r

|w j,k (x) − w j,k (y)|2d dx dy + [w j,k ]2d s,2d;B2r |x − y|n+2sd

B3r /2 \Br



Br

|v¯ j (x)|2d dx dy |x − y|n+2sd

|w j,k (x)|2d dx dy |x − y|n+2sd

+ cF j (w j,k , Br ) + cr n+2d(γ −s) H 2d   dz dz 2d ≤c  v ¯  + w j,k 2d j L 2d (B3r /2 ) L 2d (Br ) n+2sd n+2sd Rn \Br /2 |z| Rn \Br /2 |z| + cF j (w j,k , Br ) + cr n+2d(γ −s) H 2d cF j (w j,k , Br ) ≤ + cr 2d(1−s) D u˜ j 2d + cr n+2d(γ −s) H 2d L 2d (Br ) εj (3.24)

86

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for c ≡ c(datae ). Using the content of (3.22)–(3.24), we find (recall it is r ≤ 1) w j,k W 1,2d (Br ) + w j,k W s,2d (Rn ) ≤

c[F j (w j,k , Br )]1/(2d) 1/(2d)

εj

+ cD u˜ j  L 2d (Br ) + cH .

This last estimate and (3.21) imply that the sequence {w j,k }k is bounded in W 1,2d (Br , R N ) ∩ W s,2d (Rn , R N ) and therefore, up to a not relabelled subsequence, we can assume that w j,k u j weakly as k → ∞, both in W 1,2d (Br , R N ) and W s,2d (Rn , R N ), for some u j ∈ X(v¯ j , Br ). Moreover, again up to a diagonalization argument, we can also assume w j,k → u j a.e. At this point we use lower semicontinuity. Specifically, we use the convexity of z → F(x, z) and z → |z|2d to deal with the local part, and Fatou’s lemma to deal with the nonlocal one, in order to get F j (u j , Br ) ≤ lim inf F j (w j,k , Br ) , k→∞

thereby proving the minimality in (3.18) accordingly to the usual Direct Methods of the Calculus of Variations (see for instance [28]). 3.4 Step 4: Convergence to u Here we prove that, up to a non relabelled subsequence, {u j } weakly converges to the original minimizer u in W 1, p (Br , R N ). For this, we start observing that (3.20)1 guarantees that 



εj

|D v¯ j |2 + μ2

d

dx → 0 .

(3.25)

Br

Moreover, using (3.15)2 and the definition of M in (3.20)2 , we have that |v¯ j (x) − v¯ j (y)| ≤ M|x − y|γ /4 , holds for every j ∈ N and x, y ∈ Rn . In turn, this implies  Rn

 d |v¯ j (x) − v¯ j (y)|2 − M 2 |x − y|2γ +



|x − y|n+2sd

Rn

dx dy = 0

for every j ∈ N. By (3.15)3 and the definition of M0 in (3.20)2 , it is  Br

 d |v¯ j |2 − M02 dx = 0 . +

Using the information in the last two displays, recalling the definition of F j in (3.19), and also using (3.12)2 , (3.15)3 and (3.25), we find 

 d |D v¯ j |2 + μ2 dx = F(u, Br ) .

lim F j (v¯ j , Br ) = lim F(u˜ j , Br ) + lim ε j

j→∞

j→∞

Reprinted from the journal

j→∞

87

Br

C. De Filippis, G. Mingione

Minimality of u j , i.e., F j (u j , Br ) ≤ F j (v¯ j , Br ), and the above display, then give lim sup F j (u j , Br ) ≤ F(u, Br ) ,

(3.26)

j→∞

and therefore, up to relabelling, we can assume that F j (u j , Br ) ≤ F(u, Br ) + 1 holds for every j ∈ N. This and (1.11)1 imply 



ν

 d |Du j |2 + μ2 dx ≤ F j (u j , Br ) ≤ F(u, Br ) + 1 .

|Du j | p dx + ε j Br

Br

(3.27) Poincaré inequality and (3.12)3,4 yield u j  L p (Br ) ≤ u j − u˜ j  L p (Br ) + u˜ j  L p (Br ) ≤ cr Du j  L p (Br ) + cr D u˜ j  L p (Br ) + cr n/ p+γ H ≤ c[F(u, Br ) + 1]1/ p + cr n/ p H .

(3.28)

Therefore, up to a non relabelled sequence, we can assume that 1, p

u j v in W 1, p (Br , R N ), for some v such that v ∈ u + W0 (Br , R N ) . (3.29) inuity and (3.26) then imply F(v, Br ) ≤ lim inf F(u j , Br ) ≤ lim sup F j (u j , Br ) ≤ F(u, Br ) , j→∞

(3.30)

j→∞

1, p

In turn, as u − v ∈ W0 (Br , R N ), the minimality of u renders that F(u, Br ) ≤ F(v, Br ) and so F(u, Br ) = F(v, Br ). The strict convexity of z → F(·, z) implied by (1.11)2 leads to u = v almost every where on Br .

(3.31)

This means that (3.30) now becomes F(u, Br ) ≤ lim inf F(u j , Br ) ≤ lim sup F j (u j , Br ) ≤ F(u, Br ) , j→∞

j→∞

so that, recalling the definition of F j in (3.19), we gain  lim

j→∞ Rn

= lim

 Rn



j→∞ Br

 d |u j (x) − u j (y)|2 − M 2 |x − y|2γ +

|x − y|n+2sd  d |u j |2 − M02 dx = 0 .

dx dy

+

88

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Up to a not relabelled subsequence, we can therefore assume that ⎧   |u (x) − u (y)|2 − M 2 |x − y|2γ d ⎪ j j ⎪ + ⎪ ⎪ dx dy ≤ cr n+2d(γ −s) H 2d ⎪ ⎨ Rn Rn |x − y|n+2sd ⎪  ⎪ ⎪ ⎪ ⎪ ⎩

Br

 d |u j |2 − M02 dx ≤ r n+2γ d H 2d

(3.32)

+

hold for all j ∈ N. Recalling the definition of M0 in (3.20)2 , a direct consequence of (3.32)2 is u j 2d ≤ cr n+2γ d H 2d . L 2d (B ) r

(3.33)

Then, since u j ≡ v¯ j outside Br , as u j ∈ X(v¯ j , Br ), and since in turn v j ≡ 0 outside B3r /2 , by (3.33) and (3.15)1 we get u j 2d ≤ u j 2d + v j 2d L 2d (Rn ) L 2d (B ) L 2d (B

3r /2 )

r

≤ cr n+2γ d H 2d .

(3.34)

Using (3.32)1 it now follows that 



(|u j (x) − u j (y)|2 − M 2 |x − y|2γ )d+ dx dy |x − y|n+2sd B2r B2r   dx dy + cM 2d n−2d(γ −s) |x − y| B2r B2r

[u j ]2d s,2d;B2r ≤ c

≤ cr n+2d(γ −s) H 2d + cM 2d r n+2d(γ −s) = cr n+2d(γ −s) H 2d

(3.35)

where c ≡ c(datae ). Using this last estimate, and splitting again as in (3.17), we have  [u j ]2d s,2d;Rn

=2



Rn \B2r

 =2

Rn \B2r





B2r

|u j (x)|2d dx dy + [u j ]2d s,2d;B2r |x − y|n+2sd

B3r /2 \Br



|v¯ j (x)|2d dx dy |x − y|n+2sd

|u j (x)|2d dx dy + [u j ]2d s,2d;B2r n+2sd Rn \B2r Br |x − y|   dz 2d 2d  v ¯ + [u j ]2d ≤c  + u  j j s,2d;B2r . L 2d (B3r /2 ) L 2d (Br ) n+2sd Rn \Br /2 |z| +2 

Using (3.15)1 and (3.34)–(3.35) in the above display, we conclude with n+2d(γ −s) 2d [u j ]2d H s,2d;Rn ≤ cr

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89

(3.36)

C. De Filippis, G. Mingione

for c ≡ c(datae ). Recalling that β0 = s − n/(2d) > 0 in (3.5), we get (2.8) (3.36) 2d 2d [u j ]2d ≤ cr n+2d(γ −s) H 2d , 0,β0 ;Rn = [u j ]0,β0 ;B2r ≤ c[u j ]s,2d;Rn

(3.37)

for every j ∈ N and for a constant c ≡ c(datae ) which is independent of j. From now on, we will be using the following notation: F(u, Br ) := F(u, Br ) + r n H 2d + 1 .

(3.38)

By (3.27)–(3.28) we then have (recall that 2d > p and therefore H ≤ H 2d/ p + 1) u j W 1, p (Br ) ≤ c[F(u, Br )]1/ p

(3.39)

for c ≡ c(data), while (3.36)–(3.37) and r ≤ 1 imply [u j ]s,2d;Rn ≤ c[F(u, Br )]1/(2d)

2(d−1)

[u j ]s,2d;Rn [u j ]20,β0 ;Rn ≤ cF(u, Br ) , (3.40)

and

for c ≡ c(datae ). 3.5 Step 5: The Euler–Lagrange system We adopt the short notation  d F j (x, z) := F(x, z) + ε j |z|2 + μ2 .

(3.41)

Before continuing, let us recall a basic consequence of assumptions 1.11. From (1.11)2 we infer that the partial z → F(x, z) is convex for every choice of x ∈ ; at this point from the q-upper bound in (1.11)1 and [44, Lemma 2.1], it follows that |∂z F(x, z)|  (|z|2 + 1)(q−1)/2 holds for every choice of (x, z) ∈  × R N ×n . Moreover, from Sect. 3.1, 3.1 and 3.7, it is 2d > p˜ > q, so that, recalling (3.41), we conclude with  (2d−1)/2 . |∂z F j (x, z)|  |z|2 + 1

(3.42)

We are now ready to derive the Euler–Lagrange system of the functional F j . We make use of variations of the type u j + tϕ, where ϕ ∈ W01,2d (Br , R N ) ∩ W0s,2d (Rn , R N ) has compact support in Br , and t ∈ (−1, 1). More precisely, by abuse of notation, we can confine ourselves to take any ϕ ∈ W01,2d (Br , R N ) with compact support in Br . Indeed, note that ϕ ∈ W s,2d (Br , R N ) (as W 1,2d (Br , R N ) → W t,2d (Br , R N ) for every t ∈ (0, 1), [23, Proposition 2.2]); moreover, since ϕ has compact support in Br , setting ϕ ≡ 0 outside Br , yields that ϕ ∈ W s,2d (Rn , R N ) (see for instance the standard 90

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Interpolative gap bounds for nonautonomous integrals

argument in [23, Lemma 5.1]). It follows that u j + tϕ ∈ X(v¯ j , Br ). By minimality of u j , we have dF j (u j + tϕ, Br ) = 0, dt |t=0

(3.43)

that is 



0=



∂z F j (x, Du j ) · Dϕ dx + 2d Br

Rn

d−1 +

u j · ϕ dx

 d−1 |u j (x) − u j (y)|2 − M 2 |x − y|2γ + (u j (x) − u j (y)) · (ϕ(x) − ϕ(y))





+ 2d

Br

|u j |2 − M02

|x − y|n+2sd

Rn

dx dy .

(3.44)

See also [22,39] for the standard derivation concerning the nonlocal term and for nonlocal, nonlinear equations. We also note that F j (·) has standard 2d-growth, moreover, from Sects. 3.1, 3.1 and 3.7, it is 2d > p˜ > q, so that all the integrals occurring in (3.44) are absolutely convergent and all the computations leading from (3.43) are (3.43) are legal. Now, we fix parameters 0 < ≤ τ1 < τ2 ≤ r ,

(3.45)

and set ϕ := τ−h (η2 τh u j ), with ⎧ 1 ⎪ ⎨η ∈ C0 (B(3τ2 +τ1 )/4 ) 1 B(τ2 +τ1 )/2 ≤ η ≤ 1 B(3τ2 +τ1 )/4 ⎪ ⎩ 1 |Dη|  τ2 −τ 1

(3.46)

and h ∈ Rn \ {0} is such that 0 < |h|
q, this implies Du j  L q (B ) ≤

 κ˜ c n 2d F(u, B ) + r H + 1 . r (r − )κ1

By using (3.29) and (3.31), the assertion in (1.19) now follows by lower semicontinuity with κ2 = 2κd. ˜ This concludes the proof of Theorem 1 in the case p ≥ 2. Remark 3 From the proof above and the choice of the parameters in Step 1, it is not difficult to see that when q, q → p +

min{α, 2γ } , ϑ(1 − γ )

thereby forcing p˜ to approach the same value, then d → ∞. This makes the exponent κ2 in the final estimate (1.19) blow-up. This is a typical phenomenon in regularity for ( p, q)-growth functionals, already encountered in the form of the a priori estimates found in [20,25,45]. 3.10 Step 10: Iteration and a priori estimate for 1 < p < 2 In this case we use Hölder inequality to estimate  |τh Du j | p dx B(τ2 +τ1 )/2



 p/2

≤

(|Du j (x + h)|2 + |Du j (x)|2 + μ2 )

p−2 2

|τh Du j |2 dx

B(τ2 +τ1 )/2

1− p/2



(|Du j (x + h)| + |Du j (x)| + μ ) 2

2

B(τ2 +τ1 )/2

Reprinted from the journal

99

2 p/2

dx

C. De Filippis, G. Mingione (3.39)



 p/2

≤ c

(|Du j (x + h)| + |Du j (x)| + μ ) 2

2

2 p/2

dx

[F(u, Br )]1− p/2

B(τ2 +τ1 )/2

(3.72)

≤

 c|h|α0 p/2  q F(u, Br ) + Du j  L q (Bτ ) + 1 , p 2 (τ2 − τ1 )

(3.83)

where c ≡ c(data). This is the analog of (3.73) in the case p < 2. We can therefore proceed as in the case p ≥ 2; we report some of the details in the following as different bounds are coming up. As for (3.74), Lemma 1 gives u j ∈ W 1+β/2, p (B(τ2 +τ1 )/2 , R N )∩ W s,2d (Br , R N ), for β as in (3.6), with the estimate (thanks to (3.83)) u j W 1+β/2, p (B(τ

2 +τ1 )/2

)

c u j W 1, p (Bτ ) 2 (τ2 − τ1 )n/ p+β/2   c q/ p 1/ p [F(u, B + )] +Du  q (B ) +1 r j L τ2 (τ2 − τ1 )1+(β−α0 )/2   c q/ p 1/ p [F(u, B )] + Du  + 1 , ≤ q r j L (Bτ2 ) (τ2 − τ1 )n/ p+β/2 (3.84)

≤

with c ≡ c(data). It is now the turn of Lemma 3 applied with parameters s1 = s, s2 = 1 + β/2, a = 2d and t = p to get Du j  L p˜ (Bτ

1)

≤

c [u j ]θs,2d;Bτ Du j 1−θ W β/2, p (B(τ2 +τ1 )/2 ) 2 (τ2 − τ1 )κ

(3.85)

where this time it is θ :=

β , 2(1 − s) + β

(3.86)

and where p˜ is as in (3.2), and c, κ ≡ c, κ(datae ) are derived from Lemma 3. Plugging (3.40) and (3.84) into (3.85), we again arrive at the following analog of (3.78): Du j  L p˜ (Bτ

1)

≤

1−θ c +θ [F(u, Br )] p 2d n/ p+β/2+κ (τ2 − τ1 ) q(1−θ) θ c p 2d Du j  q + [F(u, B )] r L (Bτ2 ) . (τ2 − τ1 )n/ p+β/2+κ

(3.87)

Next, keeping in mind that p˜ > q by (3.7), we can apply the interpolation inequality (3.79) in (3.87), with θ1 ∈ (0, 1) as in (3.80) (but with the different, current definition of p). ˜ As for the case p ≥ 2, using (3.79) in (3.78) yields (3.81), with (3.9 2(1 − s) qθ1 (1 − θ ) (3.80,(3.86) p(s, ˜ d, β)(q − p) < 1. = p p(s, ˜ d, β) − p p[2(1 − s) + β]

We can therefore proceed exactly as after (3.82) to arrive at (1.19) and the proof of Theorem 1 is finally complete. 100

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3.11 Step 11: Proof of Corollaries 1, 2 In both cases the proof relies on the application of Theorem 1, where we verify that L F,H ,γ (u, Br ) = 0 holds for every ball Br  , for suitable choices of H > 0. We fix Br   such that [u]0,γ ;Br > 0 (otherwise the assertion is trivial) and we start from Corollary 2, where we use [3, Theorem 4] in order to get a sequence {u˜ j } of W 1,∞ (Br )-regular maps such that       |D u˜ j | p + a(x)|D u˜ j |q dx → |Du| p + a(x)|Dw|q dx , (3.88) Br

Br

and u˜ j → u strongly in W 1, p (Br ). The sequence {u˜ j } has been constructed in [3] via mollification, i.e., u˜ j := u ∗ ρε j . Here, with 0 < δ ≤ min{dist(Br , ∂)/20} being fixed, {ε j } can be taken as a sequence such that ε j → 0, 0 < ε j ≤ δ. Moreover, {ρε } is a family of standard mollifiers generated by a smooth, non-negative and radial function ρ ∈ C0∞ (B1 ) such that ρ L 1 (Rn ) = 1, via ρε (x) = ε−n ρ(x/ε). It follows that [u˜ j ]0,γ ;Br ≤ [u]0,γ ;(1+δ)Br ,

(3.89)

for every j ∈ N. Proposition 1 then implies L F,H ,γ (u, Br ) = 0 for H = [u]0,γ ;(1+δ)Br . Therefore Theorem 1 applies, with (1.19) that translates into Du L q (B ) ≤

  κ2 c 1/ p [F(u, B )] + [u] + 1 , r 0,γ ;(1+δ)B r (r − )κ1

(3.90)

which holds whenever B  Br is concentric to Br . Finally, letting δ → 0 in the above inequality leads to (1.22) and the proof of Corollary 2 is complete. We only note that we can invoke [3, Theorem 4] as we are assuming (1.16), which is a more restrictive condition than the one considered in (1.9), which is in turn sufficient to prove (3.88), as shown in [3]. For Corollary 1, the proof is totally similar and uses the same convolution argument of Corollary 2 to find the approximating sequence {u˜ j } ⊂ W 1,∞ (Br ) such that the approximation in energy F(u˜ j , Br ) → F(u, Br ) and (3.89) hold. In this case the proof of the approximation in energy is directly based on Jensen’s inequality and the convexity of G(·). The details can be found in [25, Lemma 12]. No bound on the gap q/ p is required at this stage and (1.16) is only needed to refer to Theorem 1, i.e., to prove the a priori estimates. The rest of the proof then proceeds as the one for Corollary 2. 3.12 Step 12: Proof of Theorem 2 This follows with minor modifications from the proof of Theorem 1, that we briefly outline here. As the integrand F(·) is now x-independent and convex, we are in the situation of Corollary 1, where we can verify (1.23) with b(·) ≡ 1 and G(·) ≡ F(·). It follows that L F,H ,γ (u, Br ) = 0 with the choice H = [u]0,γ ;(1+δ)Br , where δ > 0 can be chosen arbitrarily small as in the proofs of Corollaries 1, 2. As for the part Reprinted from the journal

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C. De Filippis, G. Mingione

concerning the a priori estimates, the proof follows the one given for Theorem 1 verbatim, once replacing, everywhere, α by 1. With the current choice of H , this leads to establish (3.90), from which (1.22) finally follows letting δ → 0.

4 More on the autonomous case In the autonomous case F(x, Dw) ≡ F(Dw), considering assumptions on the Hessian of F(·) of the type in (1.6), leads to bounds that are better than (1.20). Specifically, we assume that F : R N ×n → R is locally C 2 -regular in R N ×n \ {0} and satisfies ⎧    2 2 p/2 ≤ F(z) ≤ L |z|2 + μ2 q/2 + L |z|2 + μ2 p/2 ⎪ ⎨ ν |z| + μ q/2 p/2   |z|2 + μ2 |∂zz F(z)| ≤ L |z|2 + μ2 + L |z|2 + μ2 ⎪  ⎩ ( p−2)/2 ν |z|2 + μ2 |ξ |2 ≤ ∂ 2 F(z)ξ · ξ ,

(4.1)

for every choice of z, ξ ∈ R N ×n such that |z| = 0, and for exponents 1 ≤ p ≤ q. As usual, 0 < ν ≤ 1 ≤ L are fixed ellipticity constants and μ ∈ [0, 1]. Such assumptions are for instance considered in [4,45]. Using (4.1), instead of (1.11), in the statement of Theorem 2 we can replace the right-hand side quantity in (1.20) and (1.21) by the larger p+

2γ . ϑ(1 − γ )

(4.2)

The proof can be obtained along the lines of the one for Theorem 1, using different, and actually more standard estimates in Step 6. In particular, estimates (3.69)–(3.70) can be replaced by (III)1 + (III)2 ≥

1 c



 ( p−2)/2 η2 |Du j (x + h)|2 + |Du j (x)|2 + μ2 |τh Du j |2 dx Br

c|h|2 − (τ2 − τ1 )2 −

|h|2

cε j (τ2 − τ1 )2

 

 q/2 |Du j |2 + 1 dx Bτ2

 d |Du j |2 + 1 dx ,

Bτ2

and therefore estimate (3.71) can be replaced by 

 ( p−2)/2 η2 |Du j (x + h)|2 + |Du j (x)|2 + μ2 |τh Du j |2 dx Br

≤ c(III) +

 c|h|2  q F(u, Br ) + Du j  L q (Bτ ) + 1 . 2 2 (τ2 − τ1 )

This leads to change the parameters in Step 1. Specifically, we replace everywhere the quantity min{α, 2γ }, by 2γ = min{2, 2γ } starting from (3.1), while in (3.6) we 102

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Interpolative gap bounds for nonautonomous integrals

define α0 := 2β0 < 2γ . The rest of the proof follows unaltered and leads to establish (1.19) using now the bound in (4.2). Acknowledgements This work is supported by the University of Turin via the project “Regolaritá e proprietá qualitative delle soluzioni di equazioni alle derivate parziali” and by the University of Parma via the project “Regularity, Nonlinear Potential Theory and related topics”. We thank the referee for his/her comments the led to improved presentation.

Declarations Conflict of interest The authors declare they do not have any conflict of interest.

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Interpolative gap bounds for nonautonomous integrals 50. Maz’ya, V., Shaposhnikova, T.: On the Brezis and Mironescu conjecture concerning a GagliardoNirenberg inequality for fractional Sobolev norms. J. Math. Pures Appl. (IX) 81, 877–884 (2002) 51. Milman, V.A.: Extension of functions preserving the modulus of continuity. Math. Not. 61, 193–200 (1997) 52. Mingione, G.: Regularity of minima: an invitation to the dark side of the calculus of variations. Appl. Math. 51, 355–426 (2006) 53. Mingione, G., Radulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501, article no. 125197 (2021) 54. Schäffner, M.: Higher integrability for variational integrals with non-standard growth. Calc. Var. PDE 60, 77 (2021). https://doi.org/10.1007/s00526-020-01907-1 55. Schmidt, T.: Regularity of relaxed minimizers of quasiconvex variational integrals with ( p, q)-growth. Arch. Ration. Mech. Anal. 193, 311–337 (2009) 56. Serrin, J.: The problem of Dirichlet for quasilinear elliptic differential equations with many independent variables. Philos. Trans. R. Soc. Lond. Ser. A 264, 413–496 (1969) 57. Trudinger, N.: The Dirichlet problem for nonuniformly elliptic equations. Bull. Am. Math. Soc. 73, 410–413 (1967) 58. Trudinger, N.: Harnack inequalities for nonuniformly elliptic divergence structure equations. Invent. Math. 64, 517–531 (1981) 59. Ural’tseva, N.N., Urdaletova, A.B.: The boundedness of the gradients of generalized solutions of degenerate quasilinear nonuniformly elliptic equations. Vestnik Leningrad Univ. Math. 19 (1983) (Russian) English. Tran. 16, 263–270 (1984) 60. Zhikov, V.V.: On Lavrentiev’s phenomenon. Russ. J. Math. Phys. 3, 249–269 (1995) 61. Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5, 105–116 (1997) 62. Zhikov, V.V.: Lavrentiev phenomenon and homogenization for some variational problems. C. R. Acad. Sci. Paris Sér. I Math. 316, 435–439 (1993) 63. Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Analysis and Mathematical Physics (2020) 10:67 https://doi.org/10.1007/s13324-020-00418-8

Positive Liouville theorem and asymptotic behaviour for (p, A)-Laplacian type elliptic equations with Fuchsian potentials in Morrey space Ratan Kr. Giri1 · Yehuda Pinchover1 Received: 4 July 2020 / Revised: 4 July 2020 / Accepted: 21 October 2020 / Published online: 30 October 2020 © Springer Nature Switzerland AG 2020

Abstract We study Liouville-type theorems and the asymptotic behaviour of positive solutions near an isolated singular point ζ ∈ ∂ ∪ {∞} of the quasilinear elliptic equation p−2

−div(|∇u| A

A∇u) + V |u| p−2 u = 0 in  \ {ζ },

d×d ) is a symmetric where  is a domain in Rd (d ≥ 2), and A = (ai j ) ∈ L ∞ loc (; R and locally uniformly positive definite matrix. The potential V lies in a certain local Morrey space (depending on p) and has a Fuchsian-type isolated singularity at ζ .

Keywords Fuchsian singularity · Morrey spaces · Liouville theorem · ( p, A)-Laplacian Mathematics Subject Classification Primary 35B53; Secondary 35B09 · 35J62 · 35B40

1 Introduction Let  be a domain in Rd , d ≥ 2, and consider the quasilinear elliptic partial differential equation Q(u) = Q p,A,V (u) := − p,A (u) + V |u| p−2 u = 0

in .

(1.1)

Dedicated to Volodya Maz’ya on the occasion of his 80th birthday.

B

Yehuda Pinchover [email protected] Ratan Kr. Giri [email protected]; [email protected]

1

Department of Mathematics, Technion - Israel Institute of Technology, 3200 Haifa, Israel

Chapter 5 was originally published as Giri, R. Kr. & Pinchover, Y. Analysis and Mathematical Physics (2020) 10:67. https://doi.org/10.1007/s13324-020-00418-8. Reprinted from the journal

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R. Kr. Giri ,Y. Pinchover

and let ζ ∈ {0, ∞} be a fixed isolated singular point of Q p,A,V which belongs to the ideal boundary of  (to be explained in the sequel). Here 1 < p < ∞, V is a real valued potential belonging to a certain local Morrey space, and p−2

 p,A (u) := div(|∇u| A

A∇u)

d×d ) is a symmetric and locally is the ( p, A)-Laplacian, where A = (ai j ) ∈ L ∞ loc (; R uniformly positive definite matrix, and

|ξ |2A(x) := A(x)ξ · ξ =

d 

ai j (x)ξi ξ j

x ∈  and ξ = (ξ1 , . . . , ξd ) ∈ Rd .

i, j=1

We note that (1.1) is the Euler-Lagrange equation associated to the energy functional  p Q(ϕ) = Q p,A,V (ϕ) := (|∇ϕ| A + V |ϕ| p )dx ϕ ∈ Cc∞ (). (1.2) 

The quasilinear equation (1.1) satisfies the homogeneity property of linear equations but not the additivity (therefore, such an equation is sometimes called half-linear or quasilinear elliptic equations with natural growth terms). Consequently, one expects that positive solutions of (1.1) would share some properties of positive solutions of linear elliptic equations. Indeed, criticality theory for (1.1), similar to the linear case, was established in [14–16]. In [2], Frass and Pinchover studied Liouville theorems and removable singularity theorems for positive classical solutions of (1.1) under the assumptions that A is the identity matrix, V ∈ L ∞ loc (), and V has a pointwise Fuchsian-type singularity near ζ ∈ {0, ∞}, namely, |V (x)| ≤

C |x| p

near ζ.

(1.3)

Moreover, in the same paper and in [3], the asymptotic behavior of the quotient of two positive solutions near the singular point ζ has been obtained. The results in [2,3] extend the results obtained in [13, and the references therein] for second-order linear elliptic operators (not necessarily symmetric) to the quasilinear case. We note that an affirmative answer to Problem 51 of Maz’ya’s recent paper [10] follows from [2, Theorem 1.1]. The aim of the present paper is to study Liouville-type theorems, Picard-type principles, and removable singularity theorems for positive weak solutions of (1.1), by relaxing significantly the condition on the potential V ∈ L ∞ loc (). More precisely, we enable a symmetric, locally bounded, and locally uniformly positive definite matrix A, and a potential V that lies in a certain local Morrey space and has a generalized Fuchsian-type singularity at ζ (in term of a weighted Morrey norm of V ). In fact, our local regularity assumptions on A and V are almost the weakest to keep the validity of the local Harnack inequality and the local Hölder continuity of weak solutions. 108

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The outline of the present paper is as follows. In Sect. 2 we provide a short summary on the local theory of positive solutions of (1.1) with potentials in local Morrey spaces and prove Harnack convergence principle under minimal assumptions on the sort of convergence of the coefficients of the sequence of operators. In Sect. 3 we introduce the notion of a (generalized) Fuchsian singularity for the operator Q at a point ζ , and prove a uniform Harnack inequality near such a singular point which is a key result for proving (under further assumptions) that the quotient of two positive solutions near ζ admits a limit in the wide sense. Section 4 is devoted to the asymptotic behaviour of positive ( p, A)-harmonic functions near an isolated singular point for the case where A ∈ Rd×d is a symmetric and positive definite matrix. In Sect. 5 we assume that Q has a weak Fuchsian singularity at ζ and prove that it is a sufficient condition for the validity of a positive Liouville-type theorem. Finally, Sect. 6 is devoted to the study of Liouville-type theorem in the elliptically symmetric case.

2 Preliminaries We begin with notation, some definitions and assumptions. Throughout the paper,  is a domain (i. e., a nonempty open connected set) in Rd , d ≥ 2. By Br (x0 ) and Sr (x0 ) = ∂ Br (x0 ), we denote the open ball and the sphere of radius r > 0 centered at x0 , respectively, and we set Br := Br (0), Sr := Sr (0). Denote Br∗ := Rd \ Br and (Rd )∗ := Rd \ {0}, the corresponding exterior domains. For R > 0 we denote by A R the annuls A R := {x ∈ Rd | R/2 ≤ |x| < 3R/2}, and for a domain  ⊂ Rd and R > 0, we define the dilated domain /R := {x ∈ Rd | x = R −1 y, where y ∈ }. Let f , g ∈ C() be two positive functions. The notation f g in  means that there exists positive constant C such that C −1 g(x) ≤ f (x) ≤ Cg(x)

for all x ∈ .

We write 1  2 if 2 is open and 1 is compact (proper) subset of 2 . By a compact exhaustion of a domain , we mean a sequence of smooth relatively compact domains ∞  = . Finally, throughout the paper in  such that 1 = ∅, i  i+1 , and ∪i=1 i C refers to a positive constant which may vary from line to line. We first introduce a certain class of Morrey spaces, in which the potential V of the operator Q p,A,V belongs to. Definition 2.1 (Morrey spaces) A function f ∈ L 1loc (; R) is said to belong to the q local Morrey space Mloc (; R), q ∈ [1, ∞] if for any ω  

f M q (ω) :=

sup

1

 r d/q y∈ω 0 d,

q

while for p = d, the Morrey space Mloc (d; ) consists of all those f such that for some q > d and any ω   

f M q (d;ω) :=

where ϕq (r ) := logq/d





sup y∈ω 0 0 such that for any δ > 0 and all 1, p u ∈ W0 (ω) 

p

ω

|V ||u| p dx ≤ δ ∇u L p (ω;Rd ) +

C pq/( pq−d) p

V M q ( p;ω) u L p (ω) . (2.3) δ d/( pq−d)

(ii) For any ω  ω with Lipschitz boundary, there exist 0 < C = C(d, p, q, ω , ω, δ, V M q ( p;ω) ) and δ0 such that for 0 < δ ≤ δ0 and all u ∈ W 1, p (ω )  p p |V ||u| p dx ≤ δ ∇u L p (ω ;Rd ) + C u L p (ω ) . ω

We recall the Allegretto-Piepenbrink-type theorem (see, [14, Theorem 4.3]). This theorem states that Q p,A,V (ϕ) ≥ 0 for all ϕ ∈ Cc∞ () (in short, Q p,A,V ≥ 0 in ) if and only if the equation Q p,A,V (u) = 0 possesses a positive (super)solution in . Throughout the paper, we assume that Q p,A,V (ϕ) ≥ 0 for all ϕ ∈ Cc∞ (). The above assumption implies the solvability of the Dirichlet problem in bounded subdomains (see [14] Theorem 3.10 and Proposition 5.2): Lemma 2.5 Assume that Q p,A,V ≥ 0 in . Then for any Lipschitz subdomain ω  , 0 ≤ g ∈ C(ω) and 0 ≤ f ∈ C(∂ω), there exists a nonnegative solution u ∈ W 1, p (ω) of the problem Q p,A,V (v) = g in ω, and v = f on ∂ω. Moreover, the solution u is unique if either f = 0 or f > 0 on ∂ω. We recall the local Harnack inequality of nonnegative solutions of (2.1), see for example, [8, Theorem 3.14] for the case p ≤ d and [17, Theoren 7.4.1] for the case p > d. Theorem 2.6 (Local Harnack inequality) Let A, V , satisfy Assumptions 2.3, and let ω  ω  . Then for any nonnegative solution v of (2.1) in  we have sup v ≤ C inf v , ω

ω

(2.4)

where C is a positive constant depending only on d, p, dist(ω , ∂ω), A L ∞ (ω;Rd×d ) , the ellipticity constant of A in ω, and V M q ( p;ω) but not on v. The next result concerns the Harnack convergence principle for a sequence of normalized positive solutions of equations of the form (2.1) (cf. [14, Proposition 2.11], d×d ) is fixed, and {V }∞ ⊂ M q ( p;  ) converges strongly in where A ∈ L ∞ i i=1 i loc loc (; R q q Mloc ( p; ) to V ∈ Mloc ( p; )). Proposition 2.7 (Harnack convergence principle) Let {i } be a compact exhaustion ∞ is a sequence of symmetric and locally uniformly posof . Assume that {Ai }i=1 itive definite matrices such that the local ellipticity constant does not depend on Reprinted from the journal

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R. Kr. Giri ,Y. Pinchover ∞ ⊂ L ∞ ( ; Rd×d ) converges weakly in L ∞ (; Rd×d ) to a matrix i, and {Ai }i=1 i loc loc ∞ ∞ ⊂ M q ( p;  ) converges weakly in A ∈ L loc (; Rd×d ). Assume also that {Vi }i=1 i loc q q Mloc ( p; ) to V ∈ Mloc ( p; ). For each i ≥ 1, let vi be a positive weak solution of the equation Q p,Ai ,Vi (u) = 0 in i such that vi (x0 ) = 1, where x0 is a fixed reference point in 1 . Then there exists 0 < β < 1 such that, up to a subsequence, {vi } converges weakly 1, p β in Wloc () and in Cloc () to a positive weak solution v of the equation Q p,A,V (u) = 0 in .

Proof Since the sequence {Ai } is locally uniformly elliptic and converges weakly in d×d ), it follows that that A  L∞ i L ∞ ( ;Rd×d ) ≤ C for every   , and loc (; R  hence, Ai are uniformly bounded in every    expect for a set of measure zero. By the definition of vi being a positive weak solution to Q p,Ai ,Vi (v) = 0 in i , we have   p−2 p−1 1, p |∇vi | Ai Ai ∇vi · ∇u dx + Vi vi u dx = 0 for all u ∈ W0 (i ). (2.5) i

i

Also, by elliptic regularity, vi are Hölder continuous for all i ≥ 1. Fix k ∈ N. Thus, 1, p for u ∈ C0∞ (k ), by plugging vi |u| p ∈ W0 (k ), i ≥ k, as a test function in (2.5) we get p

|∇vi | Ai u L p (k )

 ≤p

k

p−1 |∇vi | Ai |u| p−1 vi |∇u| Ai

 dx +

p

k

|Vi |vi |u| p dx.

For the first term of the right-hand side of the above equation, we apply Young’s  p−1 inequality: pab ≤ εa p +[( p −1)/ε] p−1 b p , ε ∈ (0, 1), with a = |∇vi | Ai |u| p−1 and b = vi |∇u| Ai . On the second term, we use the Morrey–Adams theorem (Theorem 2.4). Then we arrive at  p

(1 − ε) |∇vi | Ai u L p (k ) ≤ p

+δ ∇(vi u) L p (

p−1 ε

 p−1

p

vi |∇u| Ai L p (k )

p

k ;R

d)

+ C vi u L p (k ) .

Since the sequence {Ai } is locally uniformly elliptic and bounded a.e., and by the inequality p

∇(vi u) L p (

k

;Rd )

 p ≤ 2 p−1 vi ∇u L p (



p

k

;Rd )

+ u∇vi L p (

k

;Rd )

,

we obtain the following estimates valid for i ≥ k and for any u ∈ Cc∞ (k ) 

 p −p p (2.6) (1−ε)Ck −2 p−1 δCk |∇vi |u L p (k ) 

 p−1 p−1 − p p p ≤ Ck +2 p−1 δ vi |∇u| L p (k )+C(d, p, q, δ, V M q ( p;k+1 ) ) vi u L p (k ) . ε

112

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We now take an arbitrary ω   and without loss of generality we assume that x0 ∈ ω. Picking a subdomain ω   such that ω  ω , we can find k ≥ 1 such that ω  k . 2p Then we choose δ < (1 − ε)21− p Ck and specialize u ∈ Cc∞ (k ) such that supp{u} ⊂ ω , 0 ≤ u ≤ 1, u = 1 in ω and |∇u| ≤

1 in ω . dist(ω, ∂ω )

(2.7)

∞ of solutions is bounded Due to the local Harnack inequality (2.6), the sequence {vi }i=1 ∞ ∞ in L (ω). In fact, by elliptic regularity {vi }i=1 is bounded in C α (ω), where 0 < α ≤ 1. Moreover, by plugging u as in (2.7) to the inequality (2.6), we get

∇vi L p (ω;Rd ) + vi L p (ω) ≤ C(d, p, q, ε, δ, dist(ω, ∂ω ), Ck , V M q ( p;k+1 ) ), p

p

∞ is bounded in W 1, p (ω). Hence up for all i ≥ k. This implies that the sequence {vi }i=1 ∞ ∞ converges uniformly to a subsequence, still denoted by {vi }i=1 , we obtain that {vi }i=1 in ω, and weakly to a nonnegative function v ∈ W 1, p (ω) ∩ C α (ω) with v(x0 ) = 1. So, we have

vi → v

uniformly in ω,

and ∇vi ∇v in L p (ω; Rd ).

We now show that v is a weak solution of Q p,A,V (u) = 0 in ω˜  ω such that x0 ∈ ω. ˜ Using the uniform convergence in ω of vi to v, we obtain      (Vi v p−1 ϕ − Vv p−1 ϕ) dx  i   ω      p−1  ≤ C vi − v L ∞ (ω) |Vi | dx +  (Vi − V)v ϕ dx  . ω

ω

q

Since the sequence {Vi } converges weakly to V in Mloc ( p; ), it is bounded in L 1loc (), therefore, the first term tends to zero, while the second term tends to zero by the weak convergence of {Vi } to V. Hence, 

p−1

ω

Vi vi

 ϕ dx →

ω

Vv p−1 ϕ dx

for all ϕ ∈ Cc∞ (ω).

(2.8)

It remains to show that p−2

p−2



ξi := |∇vi | Ai Ai ∇vi i→∞ |∇v|A A∇v := ξ in L p (ω; ˜ Rd ).

(2.9)

To prove this claim, it is enough to prove that ξi → ξ a.e., and that {ξi } is bounded in   ˜ Rd ) (see, [9] and [5, Lemma 3.73]). The boundedness of {ξi } in L p (ω; ˜ Rd ) L p (ω; p d ˜ R ). So, we need to prove clearly follows from the boundedness of {∇vi } in L (ω; the a.e. convergence of {ξi } to ξ . ∞ ⊂ L ∞ ( ; Rd×d ) converges weakly in L ∞ (; Rd×d ) By our assumption, {Ai }i=1 i loc loc d×d ). Then let us consider u as in (2.7) but with ω and ω to a matrix A ∈ L ∞ loc (; R Reprinted from the journal

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R. Kr. Giri ,Y. Pinchover

replaced by ω˜ and ω, respectively. So, by plugging u(vi − v) as a test function in (2.5), we obtain  uξi · ∇(vi − v)dx   p−1 Vi vi u(vi − v) dx. = − (vi − v)ξi · ∇u dx −

ω

ω

ω

(2.10)

For the first integral on the right-hand side of (2.10), apply Hölder’s inequality to get      p/ p  − (vi − v)ξi · ∇u dx  ≤ C p/ p (vi − v)∇u p L (ω;Rd ) ∇vi L p (ω;Rd ) ω   ω

p/ p 

≤ C( p, Cω , dist(ω, ˜ ∂ω)) (vi − v) L p (ω) ∇vi L p (ω;Rd ) →i→∞ 0, since ∇vi L p (ω;Rd ) are uniformly bounded and vi → v in L p (ω). A similar argument leading to (2.8) implies that the second integral on the right-hand side of (2.10) also converges to 0. Thus,  ω

uξi · ∇(vi − v)dx →i→∞ 0.

(2.11)

Notice that p−2

p−2

(ξi − ξ ) · (∇vi − ∇v) = (|∇vi | Ai Ai ∇vi − |∇v| Ai Ai ∇v) · (∇vi − ∇v) p−2

p−2

p−2

p−2

+(|∇v| Ai Ai∇v−|∇v|A A∇v)·(∇vi −∇v) ≥ (|∇v| Ai Ai∇v−|∇v|A A∇v)·(∇vi−∇v). d×d ), it follows [11, Proposition 2.9] Since Ai converges weakly to A in L ∞ loc (, R that Ai → A a.e.. Therefore, p−2

p−2

|∇v(x)| Ai (x) Ai (x)∇v(x) → |∇v(x)|A(x) A(x)∇v(x) for a.e. in ω, and also  p     p−2 p−2 p p |∇v| Ai Ai ∇v − |∇v|A A∇v  ≤ 2 p −1 (|∇v| Ai + |∇v|A ) ≤ C|∇v| p , since the sequence {Ai } is bounded a.e in ω. Thus, the dominated convergence theorem implies lim

   p   p−2 p−2 |∇v| Ai Ai ∇v − |∇v|A A∇v  dx = 0.

i→∞ ω

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Therefore, the Hölder inequality and the boundedness of ∇vi in L p (ω; Rd ) implies      u(|∇v| p−2 Ai ∇v − |∇v| p−2 A∇v) · (∇vi − ∇v)dx  →i→∞ 0. Ai A   ω

(2.12)

Now by using above vectors inequality we get 

p−2 p−2 (ξi − ξ ) · (∇vi − ∇v) − (|∇v| Ai Ai ∇v − |∇v|A A∇v) · (∇vi − ∇v) dx ω˜

p−2 p−2 ≤ u (ξi − ξ ) · (∇vi − ∇v)−(|∇v| Ai Ai ∇v − |∇v|A A∇v) · (∇vi − ∇v) dx  ω p−2 p−2 = u(ξi − ξ )·(∇vi − ∇v)dx − u(|∇v| Ai Ai ∇v − |∇v|A A∇v)

0≤

ω

ω

· (∇vi − ∇v)dx →i→∞ 0, where we have used (2.11), (2.12) and ∇vi ∇v in L p (ω; Rd ). It follows that 

p−2

p−2

(|∇vi | Ai Ai ∇vi − |∇v| Ai Ai ∇v) · (∇vi − ∇v) dx 

p−2 p−2 = lim (ξi − ξ ) · (∇vi −∇v) − (|∇v| Ai Ai ∇v−|∇v|A A∇v) · (∇vi − ∇v) dx = 0.

lim

i→∞ ω˜

i→∞ ω˜

(2.13)

To prove the claim (2.9), we proceed as in [5, Lemma 3.73]. Denote p−2

p−2

Di = (|∇vi | Ai Ai ∇vi − |∇v| Ai Ai ∇v) · (∇vi − ∇v). ˜ Extracting Since Di is a nonnegative function, (2.13) implies that Di → 0 in L 1 (ω). ˜ Therefore, there exists a subset Z of ω˜ of a subsequence we have Di → 0 a.e. in ω. zero measure such that for x ∈ ω˜ \ Z we have Di (x) → 0. Fix x ∈ ω˜ \ Z . Without loss of generality, we may assume that |∇v(x)| < ∞. Since Ai are locally uniformly elliptic and bounded, we have p

p

Di (x) ≥ |∇vi (x)| Ai + |∇v(x)| Ai p−1

− (|∇v(x)| Ai |∇vi (x)| Ai p

p−1

+ |∇vi (x)| Ai |∇v(x)| Ai )

−p

≥ Cω˜ |∇vi (x)| p − Cω˜ (|∇v(x)||∇vi (x)| p−1 + |∇vi (x)||∇v(x)| p−1 ) p

≥ Cω˜ |∇vi (x)| p − C(|∇vi (x)| p−1 + |∇vi (x)|), −p

where C = max(|∇v(x)|, |∇v(x)| p−1 )Cω˜ . From the above inequality, it readily follows that |∇vi (x)| is uniformly bounded with respect to i, since Di (x) → 0. Reprinted from the journal

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Let η be a limit point of ∇vi (x). Then |η| < ∞ and p−2

p−2

0 = lim (|∇vi (x)| Ai Ai ∇vi (x) − |∇v(x)| Ai Ai ∇v(x)) · (∇vi (x) − ∇v(x)) =

i→∞ p−2 (|η|A Aη

p−2

− |∇v(x)|A A∇v(x)) · (η − ∇v(x)).

This implies that η = ∇v(x). Thus we get ∇vi (x) → ∇v(x) for every x ∈ ω˜ \ Z , p−2 p−2 i.e., ∇vi → ∇v a.e. in ω˜ and |∇vi (x)| Ai Ai ∇vi (x) → |∇v(x)|A A∇v(x) a.e. in 

p−2

ω. ˜ Recall that the L p -norm of {|∇vi | Ai Ai ∇vi } is bounded in ω, ˜ therefore, (2.9) follows.   Finally, we formulate a weak comparison principle (WCP) for the case A ∈ d×d ), and V ∈ M q ( p; ). For the proof see Theorem 5.3 in [14]. L∞ loc loc (; R Theorem 2.8 (Weak comparison principle) Let ω   be a bounded Lipschitz domain, and let A and V satisfy Assumptions 2.3. Assume that the equation Q p,A,V (u) = 0 1, p admits a positive solution in Wloc () and suppose that u 1 , u 2 ∈ W 1, p (ω) ∩ C(ω) ¯ satisfy the following inequalities   Q p,A,V (u 1 ) ≤ Q p,A,V (u 2 ) in ω, Q p,A,V (u 2 ) ≥ 0 in ω, and u 2 > 0 on ∂ω, u 1 ≤ u 2 on ∂ω. Then u 1 ≤ u 2 in ω.

3 Fuchsian-type singularity We introduce the notion of Fuchsian-type singularity with respect to the equation Q p,A,V (u) = 0. We allow the domain  to be unbounded and with nonsmooth boundary, and the singular point to be ζ = ∞. Thus, it is convenient to consider the one-point ˆ d := Rd ∪ {∞}. By , ˆ d of Rd , i.e., R ˆ we denote the closure of  compactification R d ˆ in R . This should not be confused with the one-point compactification of a domain  which is also considered in the sequel. In the latter topology, a neighbourhood of infinity in  is a set of the form  \ K , where K  . ˆ is either 0 or ∞, and Throughout this paper, we assume that singular point ζ ∈ ∂  ˆ that ζ is an isolated component of ∂ . With some abuse of notation, we write  a → 0 in R and ζ = 0, a → ζ if a → ∞ in R and ζ = ∞. We extend the definition of pointwise Fuchsian-type singularity (see (1.3)). Definition 3.1 (Fuchsian singularity) Let  be a domain in Rd , and A and V satisfy ˆ be an isolated point of ∂ , ˆ where ζ = 0 or ζ = ∞. We Assumptions 2.3. Let ζ ∈ ∂  say that the operator Q p,A,V has a Fuchsian-type singularity at ζ (in short, Fuchsian singularity at ζ ) if the following two conditions are satisfied: 116

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(1) The matrix A is bounded and uniformly elliptic in a punctured neighbourhood  ⊂  of ζ , that is, there is a positive constant C such that, C −1 |ξ |2 ≤ |ξ |2A ≤ C −1 |ξ |2

∀x ∈  and ξ = (ξ1 , . . . , ξd ) ∈ Rd . (3.1)

(2) There exists a positive constant C and R0 > 0 such that 

|x| p−d/q V M q ( p;A R ) ≤ C if p = d, if p = d,

V M q (d;A R ) ≤ C

(3.2)

for all 0 < R < R0 if ζ = 0, and R > R0 if ζ = ∞, where A R := {x | R/2 ≤ |x| < 3R/2}. Definition 3.2 A set A ⊂  is called an essential set with respect to the singular point ζ if there exist real numbers 0 < a < 1 < b < ∞, and a sequence of positive numbers {Rn } with Rn → ζ such that A = ∪An , where An = {x ∈  | a Rn < |x| < b Rn }. Remark 3.3 Similar to the linear case [13], it turns out that it is sufficient to assume that inequality (3.2) is satisfied only on some essential subset of a neighbourhood of ζ . More precisely, instead of (3.2), we may assume that for some essential set A ⊂  with respect to ζ 

|x| p−d/q V M q ( p;An ) ≤ C if p = d, if p = d,

V M q (d;An ) ≤ C

where C is independent of n. (3.3)

Example 3.4 Let  = Rd \ {0} and fix 1 < p < ∞. Consider the p-Laplace operator with the Hardy potential V (x) = λ|x|− p − pu − λ

|u| p−2 u = 0 in , |x| p

(3.4)

where λ ≤ C H := |( p −d)/ p| p is the Hardy constant. A straightforward computation shows that (3.4) has Fuchsian singularity both at ζ = 0 and ζ = ∞ (in the sense of Definition 3.1). The above example implies: Example 3.5 Let  = Rd \ {0} and fix 1 < p < ∞. Consider the operator Q p,A,V = − p,A (u)+V |u| p−2 u, and assume that the matrix A satisfies (3.1) in , and |V (x)| ≤ C|x|− p in . Then Q p,A,V has Fuchsian singularity at the origin and at infinity. We now present a dilation process which uses the quasi-invariance of Fuchsian equations of the form (2.1) under the scaling x → Rx for R > 0. Let A R and V R be the scaled matrix and potential defined by A R (x) := A(Rx), V R (x) := R p V (Rx) Reprinted from the journal

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Consider the annular set A R = (B3R/2 \ B¯ R/2 ) ∩ . By our assumption that ζ is an isolated singular point it follows that A R /R is fixed annular set A˜ for R ‘near’ ζ , and for such R we have

V R M q ( p;A˜ ) = V R M q ( p;A R /R) = R p−d/q V M q ( p;A R ) ≤ C,

(3.5)

for p = d and while for p = d,

V R M q (d;A˜ ) = V R M q (d;A R /R) = V M q (d;A R ) ≤ C.

(3.6)

Let Y := lim /Rn , where Rn → ζ . Note that since by our assumptions, ζ is an n→∞

ˆ it follows that Y = (Rd )∗ = Rd \ {0}. isolated component of ∂ , The limiting dilation process is defined as follows. Let ζ = 0 or ζ = ∞, and assume that there is a sequence {Rn } of positive numbers satisfying Rn → ζ such that 

n→∞

A Rn −−−→ A

d×d ), and in the weak topology of L ∞ loc (Y ; R

V Rn −−−→ V

in the weak topology of Mloc ( p; Y ).

n→∞

q

(3.7)

Motivated by the Harnack convergence principle (see, Proposition 2.7), we define the limiting dilated equation with respect to equation (2.1) and the sequence {Rn } that satisfies (3.7) by D{Rn } (Q)(w) := − p,A (w) + V|w| p−2 w = 0 on Y .

(3.8)

The following proposition establishes a key invariance property of the limiting dilation process. Proposition 3.6 Let A, V , satisfy Assumptions 2.3. Suppose that the quasilinear equaˆ and let D{Rn } (Q)(w) be tion (2.1) has a Fuchsian singularity at ζ ∈ {0, ∞} ⊂ ∂ , a limiting dilated operator corresponding to a sequence Rn → ζ . Then the equation D{Rn } (Q)(w) = 0 in Y has Fuchsian singularity at ζ . Proof It is trivial to verify that the proposition holds true when p = d. Now for p = d, by Remark 3.3, there exists C > 0 and an essential set A = ∪An , where An = {x ∈  | a Rn < |x| < b Rn } such that

|x| p−d/q V M q ( p;An ) ≤ C

∀n ∈ N.

We claim that

|x| p−d/q V M q ( p;A/Rn ) ≤ C

∀n ∈ N.

Recall that for each n, An /Rn is a fixed annular set A˜ = {x | a < |x| < b}. Assume that p < d, so, p − d/q > 0. Then we have

|x| p−d/q V M q ( p;An /Rn ) ≤ b p−d/q V M q ( p;A˜ ) 118

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Positive Liouville theorem and asymptotic behaviour for…

≤ b p−d/q lim inf V Rn M q ( p;An /Rn ) n→∞

p−d/q

=b lim inf Rn V M q ( p;An ) n→∞   p−d/q b ≤ lim inf |x| p−d/q V M q ( p;An ) ≤ C. n→∞ a p−d/q

q

For p > d, Mloc () = L 1loc (), and similarly we get |x| p−d V L 1 (An /Rn ) ≤ C.

 

Definition 3.7 Let Gζ be the germs of all positive solutions u of the equation Q p,A,V (w) = 0 in relative punctured neighbourhoods of ζ . We say that ζ is a regular point of the above equation if for any two positive solutions u, v ∈ Gζ lim

u(x)

x→ζ v(x) x∈

exists in the wide sense.

Next, we define two types of positive solutions of minimal growth, the first one was introduced by Agmon [1] for the linear case, and was later extended to p-Laplacian type equations in [2,15,16]. Definition 3.8 (1) Let K 0 be a compact subset of . A positive solution u of the equation Q p,A,V (u) = 0 in  \ K 0 is said to be a positive solution of minimal growth in a neighbourhood of infinity in  if for any smooth compact subset K of  with K 0  int K and any positive supersolution v ∈ C( \ int K ) of the equation Q p,A,V (u) = 0 in  \ K , we have u≤v

on ∂ K

⇒

u≤v

in  \ K .

(2) A positive solution of the equation Q p,A,V (u) = 0 in  which has minimal growth in a neighbourhood of infinity in  is called a ground state of Q p,A,V in . ˆ and let u be a positive solution of the equation Q p,A,V (u) = 0 in . (3) Let ζ ∈ ∂  ˆ Then u is said to be a positive solution of minimal growth in a neighbourhood of ∂ \{ζ } ˆ of ζ such that  := ∂ K ∩  is smooth, and if for any relative neighbourhood K   any positive supersolution v ∈ C(( \ K ) ∪ ) of the equation Q p,A,V (u) = 0 in  \ K , we have u≤v

on 

⇒

u≤v

in  \ K .

Proposition 3.9 Suppose that Q p,A,V ≥ 0 in , and Q p,A,V has an isolated Fuchsian ˆ where ζ ∈ {0, ∞}. Then equation (2.1) admits a positive singularity at ζ ∈ ∂ , ˆ \ {ζ }. solution in  of minimal growth in a neighbourhood of ∂  Proof Let ζ = 0. By [14, Theorem 5.7], for any x0 ∈ , the equation Q p,A,V (u) = 0 admits a positive solution u x0 of the equation Q p,A,V (u) = 0 in  \ {x0 } of minimal growth in a neighbourhood of infinity in . Note that the proof of [14, Theorem 5.7] Reprinted from the journal

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R. Kr. Giri ,Y. Pinchover

applies also in case that x0 ∈ ∂ is an isolated singular point of ∂. Hence, (2.1) ˆ \ {0}. admits a positive solution in  of minimal growth in a neighbourhood of ∂  Now consider the case when ζ = ∞. Let {xn } ⊂  be a sequence such that xn → ∞. Fix a reference point x0 ∈ , and a compact smooth exhaustion {k } of . For n ∈ N, denote by u xn a positive solution of the equation Q p,A,V (u) = 0 in  \ {xn } of minimal growth in a neighbourhood of infinity in , and let vxn (x) := u xn (x)/u xn (x0 ). By the Harnack convergence principle, up to a subsequence, vxn converges locally uniformly to v which is a positive solution of the equation Q p,A,V (u) = 0 in . We claim that v is a positive solution in  of minimal growth in a neighbourhood ˆ \ {∞}. Indeed, let K ⊂  ˆ be a punctured neighborhood of ∞ with smooth of ∂  boundary, and let w be a positive supersolution of the equation Q p,A,V (u) = 0 in  =  \ K , such that v ≤ w on ∂ K . Then for any ε > 0 there exists n ε such that vxn ≤ (1 + ε)w on ∂ K for all n ≥ n ε . Recall that by the construction of vxn , for a fixed n we have, vxn = limk→∞ vn,k , where vn,k restricted to k ∩  is a positive solution of the equation Q p,A,V (u) = 0 which vanishes on ∂k ∩  . Therefore, by the weak convergence principle, vxn ≤ (1 + ε)w in k ∩  , for all n ≥ n ε . and therefore, v ≤ (1 + ε)w in  . Since ε is arbitrarily small, we have v ≤ w in  . Thus, v is a positive solution in  of minimal ˆ \ {ζ }. growth in a neighbourhood of ∂    We extend Conjecture 1.1 in [2] to the more general setting considered in the present paper. Our main goal is to prove it under some further relatively mild assumptions. Conjecture 3.10 Assume that Eq. (2.1) has a Fuchsian-type isolated singularity at ˆ and admits a (global) positive solution. Then ζ ∈ ∂ (i) ζ is a regular point of Eq. (2.1). (ii) Equation (2.1) admits a unique (global) positive solution of minimal growth in a ˆ \ {ζ }. neighborhood of ∂  Next, we recall the notions of subcriticality and criticality (for more details see [14]). Definition 3.11 Assume that Q p,A,V ≥ 0 in . Then Q p,A,V is called subcritical in q  if there exists a nonzero nonnegative function W ∈ Mloc ( p; ) such that  Q p,A,V (ϕ) ≥



W |ϕ| p dx

for all ϕ ∈ Cc∞ ().

(3.9)

If this is not the case, then Q p,A,V is called critical in . Theorem 3.12 [14] Let Q p,A,V ≥ 0 in . Then the following assertions are equivalent: (i) Q p,A,V is critical in . (ii) The equation Q p,A,V (u) = 0 in  admits a unique positive supersolution. (iii) The equation Q p,A,V (u) = 0 in  admits a ground state φ. The following uniform Harnack inequality near an isolated Fuchsian singular point ζ is a key ingredient for obtaining regularity results at ζ . 120

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Theorem 3.13 (Uniform Harnack inequality) Let A and V satisfy Assumptions 2.3, ˆ where and assume that Q = Q p,A,V has an isolated Fuchsian singularity at ζ ∈ ∂ , ¯ ζ = 0 or ζ = ∞. Let u, v ∈ Gζ . Consider the annular set Ar := (B3r /2 \ Br /2 ) ∩  , where  is a punctured neighbourhood of ζ . Denote ar := inf

x∈Ar

u(x) , v(x)

u(x) . x∈Ar v(x)

Ar := sup

Then there exists C > 0 independent of r , u and v such that Ar ≤ Car

∀ r near ζ.

Proof Fix positive solutions u and v in  ⊂ , a fixed punctured neighbourhood of ζ . For r > 0, let us consider the annular set A˜r := (B2r \ B¯ r /4 ) ∩  . Since ζ = 0 (respectively, ζ = ∞) is an isolated singular point, then for r < r0 (respectively, ˜ r > r0 ) Ar /r and A˜r /r are fixed annulus A and A˜ respectively and A  A. Now for such r , we define u r (x) := u(r x) for x ∈  /r . Then the function u r is a positive solution of the equation Q r [u r ] := − p,Ar (u r ) + Vr (x)|u r | p−2 u r = 0

˜ in A,

(3.10)

where Ar (x) := A(r x) and Vr = r p V (r x). Similarly, vr (x) := v(r x) for x ∈  /r ˜ In light of estimates (3.5) and (3.6), the norms Vr q satisfies Q r [vr ] = 0 in A. M ( p;A˜ ) ˜ Also, by (3.1), the matrices Ar (x) of the scaled potentials are uniformly bounded A. ˜ Therefore, the local Harnack are uniformly bounded and uniformly elliptic in A. inequality (Theorem 2.6) in the annular domain A˜ implies u(x) u r (x) u r (x) u(x) = sup ≤ C inf = C inf = Car , x∈A vr (x) x∈Ar v(x) x∈Ar v(x) x∈A vr (x)

Ar = sup

where the constant C is independent of u and v and r for r near ζ .

 

The weak comparison principle (Theorem 2.8) implies the following monotonicity useful result: Lemma 3.14 Let A and V satisfy Assumptions 2.3, and assume that u, v ∈ Gζ are defined in a punctured neighbourhood  of ζ . For r > 0, denote m r := inf  Sr ∩

u(x) , v(x)

Mr := sup

Sr ∩

u(x) . v(x)

(3.11)

(i) The functions m r and Mr are finally monotone as r → ζ . Specifically, there are numbers 0 ≤ m ≤ M ≤ ∞ such that m := lim m r , and M := lim Mr . r →ζ

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r →ζ

121

(3.12)

R. Kr. Giri ,Y. Pinchover

(ii) Suppose further that u and v are both positive solutions of (2.1) in  of minimal ˆ \ {ζ }, then 0 < m ≤ M < ∞ and m r  m and Mr  M when growth in ∂  r → ζ. The proof of Lemma 3.14 follows the same steps as in [2, Lemma 4.2] (where A is the identity matrix and V ∈ L ∞ loc ()), and therefore it is omitted. Remark 3.15 Let A ∈ Rd×d be a symmetric, positive definite matrix. Then clearly, Lemma 3.14 also holds if the sphere Sr is replaced by the set ∂ E A (r ) = {x ∈ Rd | |x|A−1 = r }. The following result readily follows from the second part of Lemma 3.14. ˆ Corollary 3.16 Suppose that (2.1) has a Fuchsian isolated singularity at ζ ∈ ∂ . Let u, v be two positive solutions of (2.1) of minimal growth in a neighbourhood of ˆ \ {ζ }. Then ∂ mv(x) ≤ u(x) ≤ Mv(x) x ∈ , where 0 < m ≤ M < ∞ are defined in (3.12). As in [2], the regularity at ζ implies a positive Liouville-type theorem. Proposition 3.17 Suppose that Q p,A,V has a regular and isolated Fuchsian singularity ˆ Then Eq. (2.1) admits a unique positive solution in  of minimal growth at ζ ∈ ∂ . ˆ \ {ζ }. in a neighbourhood of ∂  Proof Existence: Follows from Proposition 3.9. Uniqueness: Let u and v be two solutions of (2.1) of minimal growth in a neighˆ \ {ζ }. Then by Corollary 3.16, we have bourhood of ∂  mv(x) ≤ u(x) ≤ Mv(x) x ∈ , where 0 < m ≤ M < ∞ are defined in (3.11) and (3.12). In addition, since ζ is a regular point, it follows that lim

x→ζ x∈

u(x) v(x)

exists and is positive.

Thus, we have m = M and u(x) = Mv(x).

 

The following proposition asserts that the regularity of a Fuchsian singular point with respect to a limiting dilated equation implies the regularity of the corresponding singular point for the original Eq. (2.1). The proposition extends Proposition 2.2 in [2], where A is the identity matrix and V ∈ L ∞ loc () satisfies (1.3). As in [2], the proof of below relies upon the Harnack convergence principle, the WCP and the uniform Harnack inequality. 122

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Proposition 3.18 Let A, V , satisfy Assumptions 2.3. Suppose that the operator Q = ˆ and there is a sequence Q p,A,V has an isolated Fuchsian singularity at ζ ∈ ∂ , Rn → ζ , such that either 0 or ∞ is a regular point of a limiting dilated equation D{Rn } (Q)(w) = 0 in . Then ζ is a regular point of the equation Q(u) = 0 in . Proof Let u, v ∈ Gζ and set m r := inf  Sr ∩

u(x) , v(x)

u(x) , Sr ∩ v(x)

Mr := sup

(3.13)

where  is a punctured neighbourhood of ζ . By Lemma 3.14, M := limr →ζ Mr and m := limr →ζ m r exist in the wide sense, and we need to prove that M = m. Now if M := limr →ζ Mr = ∞ (respectively, m := limr →ζ m r = 0), then by the uniform Harnack inequality, Lemma 3.13, we have m = ∞ (respectively, M = 0), and hence the limit lim

u(x)

x→ζ v(x) x∈

exists in the wide sense.

Thus, we may assume that u v in some neighbourhood  ⊂  of ζ . Fix x0 ∈ Rd such that Rn x0 ∈  for all n ∈ N and define

u n (x) :=

u(Rn x) v(Rn x) , vn (x) := . u(Rn x0 ) u(Rn x0 )

(3.14)

Then by the definition of the set Gζ , u n and vn are positive solutions of the equation −  p,An (w) + Vn (x)|w| p−2 w = 0 in  /Rn ,

(3.15)

where An (x) := A Rn (x) and Vn (x) := V Rn (x), are the associated scaled matrix and potential, respectively. Since u n (x0 ) = 1 and vn (x0 ) 1, the Harnack convergence principle (Proposition 2.7) implies that {Rn } admits a subsequence (still denoted by {Rn }) such that lim u n (x) := u ∞ (x), and lim vn (x) := v∞ (x)

n→∞

n→∞

(3.16)

locally uniformly in Y = lim  /Rn , and u ∞ and v∞ are positive solutions of the n→∞ limiting dilated equation D{Rn } (Q)(w) = − p,A (w) + V|w| p−2 w = 0 on Y . Thus, for any fixed R > 0 we have u ∞ (x) u n (x) u n (x) = sup lim = lim sup n→∞ n→∞ vn (x) x∈S R v∞ (x) x∈S R x∈S R vn (x) sup

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(3.17)

R. Kr. Giri ,Y. Pinchover

u(Rn x) u(Rn x) = lim = lim M R Rn = M, sup n→∞ n→∞ v(Rn x) Rn x∈S R Rn v(Rn x) R

= lim sup n→∞ x∈S

where we have used the existence of limr →ζ Mr = M, and the local uniform conver∞ (x) gence of the sequence {u n /vn } in Y . Similarly, we have inf uv∞ (x) = m. Now by our x∈S R

assumption, either ζ1 = 0 or ζ1 = ∞ is a regular point of the dilated equation (3.17), so the limit lim

u ∞ (x)

exists.

x→ζ1 v∞ (x) x∈Y

Hence, m = M, which implies that lim

u(x)

x→ζ v(x) x∈

exists.

Thus, ζ is a regular point of the equation Q(u) = 0 in .

 

4 Asymptotic behaviour of (p, A)-harmonic functions In this section, we study the regularity of positive ( p, A)-harmonic functions at ζ , when A ∈ Rd×d is a fixed symmetric and positive definite matrix. We prove the following theorem. Theorem 4.1 Assume that 1 < p < ∞ and d ≥ 2. Let A ∈ Rd×d be a fixed symmetric and positive definite matrix. Then (i) for p ≤ d, ζ = 0 is a regular point of the equation − p,A (u) = 0 in Rd \ {0}. (ii) for p ≥ d, ζ = ∞ is a regular point of the equation − p,A (u) = 0 in Rd . The proof of Theorem 4.1 depends on the asymptotic behaviour of positive solutions near an isolated singularity. Before proving Theorem 4.1, we first establish the existence of a ‘fundamental solution’ (given by an explicit form) for the ( p, A)-Laplacian p−2

− p,A (u) = div(|∇u|A A∇u), where A ∈ Rd×d is a fixed symmetric and positive definite matrix. Lemma 4.2 (Fundamental solution) Let A ∈ Rd×d be a fixed symmetric, positive definite matrix, and let A−1 be its inverse matrix. Fix y ∈ Rd . Let  ( p−d)/( p−1) x ∈ Rd , p = d, |x − y|A−1 μ(x − y) := C p,d,A x ∈ Rd , p = d, − log |x − y|A−1 124

(4.1)

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Positive Liouville theorem and asymptotic behaviour for…

where

C p,d,A

⎧ ⎨ p − 1 (|A|1/2 ω )−1/( p−1) p = d, d := d − p ⎩ p = d, (|A|1/2 ωd )−1/(d−1)

|A| is the determinant of A, and ωd is the hypersurface area of the unit sphere in Rd . Then − p,A (μ(x − y)) = δ y (x)

in Rd .

Remarks 4.3 1. Note that μ is a positive function if and only if p < d, which implies that − p,A (where A is a constant matrix) is subcritical in Rd if and only if p < d (see Theorem 4.6). 2. In the sequel we abuse the notation and write μ(|x − y|) := μ(x − y). Proof Denote C := C p,d,A , and assume first that p = d. Without loss of generality, we may assume that y = 0. Recall that for a fixed symmetric matrix A ∈ Rd×d , the gradient of the associated quadratic form is given by ∇(Ax · x) = 2Ax.

(4.2)

Therefore, the chain rule and (4.2) implies  p−d (1−d)/( p−1)  −1 |x|A−1 ∇ (A x ·x)1/2 p−1 p−d (1−d)/( p−1) 1 =C |x| ∇(A−1 x ·x) p−1 A−1 2|x|A−1 p − d (1−d)/( p−1) 1 |x| =C A−1 x p − 1 A−1 |x|A−1 p − d (1−d)/( p−1)−1 −1 |x| =C A x. p − 1 A−1

∇μ(x) = C

So,    p − d  ((1−d)/( p−1)−1) −1 |x|  |∇μ(x)|A = |C| |A x|A p − 1  A−1    p − d  (1−d)/( p−1)  |x| == |C|  . p − 1  A−1 Consequently, (1−d)( p−2)

p−2

η(x) := |∇μ(x)|A A∇μ(x) = C|C| p−2c( p, d)|x|A−1p−1 = C|C| p−2c( p, d)|x|−d x, A−1

Reprinted from the journal

1−d

−1

|x|Ap−1 x −1 (4.3)

125

R. Kr. Giri ,Y. Pinchover

 p−2    where c( p, d) =  p−d p−1  Denote ηi (x) :=

p−d p−1 . p−2 C|C| c( p, d)|x|−d x. A−1 i

Then by (4.2)

  ∂ηi (x) −d−2 −1 = C|C| p−2c( p, d) |x|−d − d|x| (A x) x i i . A−1 A−1 ∂ xi Therefore, for all x ∈ Rd \ {0} we have − p,A (μ(x)) = −div η(x) = −C|C| p−2 c( p, d) d    −d−2  −1 d|x|−d = 0. − d|x| (A x) x i i A−1 A−1 i=1

Similarly, for p = d, we obtain that Cd,A log(|x|A−1 ) satisfies −d,A (u) = 0 in Rd \ {0}. We now find the constant C = C p,d,A ∈ R such that μ satisfies −  p,A (μ) = δ0

in Rd ,

(4.4)

in the sense of distributions. Recall that the ellipsoid E A (r ) = {x ∈ Rd | |x|A−1 < r } with ‘center’ at the origin and ‘radius’ r > 0, is affinely equivalent to the ball Br (0). Hence, E A (r ) is a relatively compact, convex subset of Rd . Let us first consider the case p = d. Note that if p < d, then lim x→0 μ(x) = ∞, but nevertheless, μ is integrable near the origin. Using the divergence theorem on the ellipsoid E A (r ), it follows that the function μ should satisfy  −1 =

E A (r )

   p−2 div |∇μ|A A∇μ dx =

p−2

∂ E A (r )

|∇μ|A A∇μ · ndS,

(4.5)

where n = A−1 x/|x|A−1 (x) is the unit outward normal to the boundary of the ellipsoid E A (r ) and dS is the hypersurface element area. Recall that by (4.3) we have p−2 |∇μ(x)|A A∇μ(x)

= C|C|

   p − d  p−2 p − d x  ,  p − 1 p − 1 |x|d −1 A

p−2 

and the hypersurface area of ∂ E A (r ) is given by r d−1 |A|1/2 ωd (see for example, [12, p. 238]), it follows from (4.5) that     p − d  p−2 p − d 1  dS −1 = C|C|  p − 1 p − 1 ∂ E A (r ) |x|d−1 A−1  p−2    p − d 1/2 p−2  p − d  |A| ωd . = C|C|  p − 1 p−1 p−2 

126

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So, for p = d, we have C = C p,d,A =

p−1 (|A|1/2 ωd )−1/( p−1) . d−p

Similarly, for p = d one obtains C(d, A) = (|A|1/2 ωd )−1/(d−1) .

 

Theorem 4.4 Let 1 < p ≤ d and A ∈ Rd×d be a fixed symmetric positive definite matrix. Suppose that u is a positive solution of the equation − p,A (v) = 0 in a punctured neighbourhood of 0 which has a non-removable singularity at 0, then u(x) ∼ μ(x), x→0

where μ is the fundamental solution of − p,A in Rd given by (4.1). Remark 4.5 For the case when A = I , Theorem 4.4 has been proved in [6, Theorem 1.1 and [7]], see also [4]. We give a slightly different proof of Theorem 4.4 by using Lemma 3.14. Proof of Theorem 4.4 Assume that 1 < p < d, the proof for the case when p = d needs only minor modifications, and therefore, it is omitted. It is known [18,19] that any positive solution v of the equation − p,A (u) = 0 in a punctured ball Br \ {0} has either a removable singularity at 0, or else, v(x) μ(x)

as x → 0.

Since u has a nonremovable singularity at 0, it follows that there exists C > 0 such that C −1 μ(x) ≤ v(x) ≤ Cμ(x) for all x in a small punctured neighbourhood of 0. Let {xn } be a sequence converging to 0. Denote rn = |xn |A−1 , and define u(x) , ∂ E A (rn ) μ(x)

Mn := sup

m n :=

u(x) . ∂ E A (rn ) μ(x) inf

Then the sequence {Mn } is bounded and bounded away from 0. Moreover, by Lemma 3.14 and Remark 3.15, the sequence {Mn } is finally monotone. Let M := limn→∞ Mn . Then   lim

n→∞

sup (u/μ − M) = 0.

∂ E A (rn )

Fix x0 ∈ Rd such that rn x0 ∈ ∗ =  \ {0} for all n ∈ N and consider the following functions u n (x) :=

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u(rn x) μ(rn x) , and μn (x) := . μ(rn x0 ) μ(rn x0 ) 127

R. Kr. Giri ,Y. Pinchover

Then u n and μn are positive solution of the equation − p,A (w) = 0 in ∗ /rn . Note d− p

p−d

p−1 that μn (x) = |x0 |Ap−1 −1 |x|A−1 with μn (x 0 ) = 1, hence, μn does not depend on n. On the other hand, u n (x0 ) 1, hence, the Harnack convergence principle implies that, up to a subsequence,

lim u n (x) = u ∞ (x)

n→∞

locally uniformly in Rd \ {0} and u ∞ is a positive solution of − p,A (w) = 0 in Rd \ {0}. Then for any fixed R > 0, as in Proposition 3.18, it follows that M=

u ∞ (x) . x∈∂ E A (R) μ(x) sup

Hence, for any R > 0, we have u ∞ (x) ≤ Mμ(x) for all x ∈ ∂ E A (R). Note that ∇μ = 0. Recall the strong comparison principle, [2, Theorem 3.2] which is proved for the case where the principal part of the operator Q is the p-Laplacian. Nevertheless, it is easy to check that the proof applies also to our setting, and in particular, for the ( p, A)operator. Hence, the strong comparison principle implies that u ∞ (x) = Mμ(x). Similarly, let m := lim m n , it follows that for any R > 0, we have mμ(x) ≤ u ∞ (x) n→∞

for all x ∈ ∂ E A (R), and consequently, u ∞ (x) = mμ(x). Therefore, M = m, and this implies that lim u/μ − M L ∞ (∂ E A (|xn |

n→∞

A−1 )

= 0.

In other words, u is almost equal to Mμ on a sequence of concentric ellipsoids converging to 0. Using the WCP in the annuli An := {|xn+1 |A−1 ≤ |x|A−1 ≤ |xn |A−1 }, n ≥ 1, it follows that lim u/μ − M L ∞ (∂ E A (r )) = 0.

r →0

Finally we note that it can be easily verified that M is independent of the choice of   the sequence {xn }. Thus, the theorem is proved. Similar to the case of the p-Laplacian in Rd we have: Theorem 4.6 Assume that A ∈ Rd×d is a symmetric, positive definite matrix. Then the operator − p,A is critical in Rd if and only if p ≥ d. Proof If p < d, then by the Hardy inequality for the p-Laplacian 

 Rd

|∇ϕ| p dx ≥

d−p p

p  Rd

|ϕ| p dx 1 + |x| p 128

∀ϕ ∈ Cc∞ (Rd ),

(4.6)

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Positive Liouville theorem and asymptotic behaviour for…

the operator − p,A is subcritical in Rd . Suppose now that p ≥ d. By Theorem 3.12, the following Dirichlet problem admits a unique positive solution wk : ⎧ ⎨  p,A (wk ) = 0 w (x) = 1 ⎩ k wk (x) = 0

in Bk \ B1 , on S1 , on Sk .

(4.7)

By the WCP, {wk }k∈N is an increasing sequence satisfying 0 ≤ wk ≤ 1, and therefore, converging to a positive solution w of  p,A (v) = 0 in Rd \ B1 , that clearly has minimal growth at infinity in Rd . Thus, it is enough to show that w = 1 in Rd \ B1 . We obviously have w ≤ 1. On the other hand, since |μ(x)| → ∞ as x → ∞, it follows that for any ε > 0, there is kε such that 1 − ε|μ(x)| ≤ wk obviously on S1 and also on Sk for every k ≥ kε . Invoking again the WCP, it follows that 1 − ε|μ| ≤ w in Bk \ B1 and it follows 1 − ε|μ| ≤ w in Rd \ B1 . By letting ε → 0, we conclude that 1 ≤ w. Thus, w = 1 in Rd \ B1 , and u 0 = 1 is a ground state. Hence, by Theorem 3.12, the   operator − p,A is critical in Rd . Corollary 4.7 Assume that 1 < p < ∞, and A ∈ Rd×d is a symmetric, positive definite matrix. Let u be a positive solution of the equation − p,A (u) = 0 in a neighbourhood of infinity. Then lim x→∞ u(x) exists in the wide sense. Moreover, if p ≥ d (resp., p < d ), then lim x→∞ u(x) = 0 (resp, lim x→∞ u(x) = ∞). Proof From Lemma 3.14 (with v = 1), it follows that the functions given by m r := inf u(x), x∈Sr

Mr := sup u(x) x∈Sr

are monotone for large enough r . If limr →∞ m r = ∞, then clearly lim x→∞ u(x) = ∞. Assume now that m = limr →∞ m r < ∞. Then for any ε > 0 the function u −m +ε is a positive solution of  p,A (w) = 0 in some neighbourhood infinity. Then by the uniform Harnack inequality (3.13), we get Mr − m + ε ≤ C(m r − m + ε), for large enough r . By taking r → ∞, we get 0 ≤ M − m ≤ (C − 1)ε. This implies that M = m < ∞, and lim u(x) = m = M < ∞. Thus, u has a finite limit as |x|→∞

x → ∞. Let p < d, and suppose that there exists a positive ( p, A)-harmonic function u in a neighbourhood such that lim x→∞ u(x) = ∞. By repeating the proof of Theorem 4.6, with u replacing μ, it would follow that − p,A is critical in Rd , a contradiction to Theorem 4.6. It remains to prove that lim u(x) = 0 if p ≥ d. By Theorem 4.6,  p,A is critical x→∞

in Rd with a ground state u 0 = 1. Since a ground state is an entire positive solution Reprinted from the journal

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of minimal growth at infinity, it follows that lim x→∞ u(x) = 0. Hence, the lemma follows.   Next we discuss the asymptotic behaviour of positive ( p, A)-harmonic functions at ∞ for p ≥ d. Theorem 4.8 Assume that p ≥ d and A ∈ Rd×d is a symmetric, positive definite matrix. Let u be a positive solution of the equation − p,A (w) = 0 in a neighbourhood of infinity in Rd . Then either u has a (finite) positive limit as x → ∞, or u(x) ∼ −μ(x), x→∞

where μ is the fundamental solution of − p,A in Rd given by (4.1). To show this theorem, we use a Kelvin-type transform (see, Definition A.1 of [2] for A = I ). Definition 4.9 For x ∈ Rd , we denote by x˜ := x/|x|2A−1 . Then x˜ is the inverse point ˜ A−1 = 1/|x|A−1 . Let u be a function with respect to the ellipsoid E A (1). In particular, |x| either defined in the ellipsoid E A (1) \ {0}, or on Rd \ E A (1). The generalized Kelvin transform of u is given by K [u](x) := u(x) ˜ = u(x/|x|2A−1 ).  For p = d, the Dirichlet integral  |∇u|dA dx is conformally invariant since λmin |∇u|d ≤ |∇u|dA ≤ λmax |∇u|d , where λmin , λmax are the lowest and greatest eigenvalues of A. The (d, A)-harmonic equation −d,A (u) = 0 is therefore, invariant under the generalized Kelvin transform. In particular, if u is (d, A)-harmonic, then K [u] is also (d, A)-harmonic (see also, Lemma 4.10). Hence, for p = d, Theorem 4.8 follows from Theorem 4.4. Lemma 4.10 Assume that p > d, and let A ∈ Rd×d be a symmetric, positive definite matrix. Set β := 2( p − d). Suppose that u is a solution of − p,A (u) = 0 in a neighbourhood of infinity (respectively, in a punctured neighbourhood of origin). Then v := K [u] is a solution of the equation   p−2 β − div B(v) := −div (|x|A−1 |∇v|A A∇v) = 0,

(4.8)

in a punctured neighbourhood of origin (respectively, in a neighbourhood of infinity). Proof Denote x˜i := xi /|x|2A−1 . By using the chain rule and (4.2), it follows that ˜ x) ˜ x) ˜ ˜ − 2(∇u( ˜ · x)A ˜ −1 x, ∇v(x) = |x| ˜ 2A−1 ∇u( where ∇˜ denotes the gradient with respect to x. ˜ Therefore, 130

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  ˜ x)−2( ˜ x)· |∇v(x)|2A = A|x| ˜ 2A−1 ∇u( ˜ ∇u( ˜ x) ˜ x˜ ·  2  ˜ x)| ˜ x)−2( ˜ x)· |x| ˜ −1 ∇u( ˜ 2 |x| ˜ 4 −1 . ˜ ∇u( ˜ x)A ˜ −1 x˜ = |∇u( A

A

A

˜ x)| ˜ A |x| ˜ 2A−1 . Thus, |∇v(x)|A = |∇u( β

p−2

Consider B(v) = |x|A−1 |∇v|A A∇v, where β = 2( p − d). Following the same steps of the computation in [2, Lemma A.1], we conclude that   p−2 β ˜  div B(v) = div(|x|A−1 |∇v|A A∇v) = |x| ˜ 2d (u(x)) ˜ = 0. A−1 p,A   ( p−d)/( p−1) |x|A−1

is a positive ( p, A)-harmonic function  (d− p)/( p−1)  ) = 0 in the in the punctured space. Lemma 4.10 implies that div B(|x|A−1 punctured space. Remark 4.11 By Lemma 4.2,

The following two lemmas are the analogous results for the p-Laplacian proved in [2, Appendix A]. For the completeness, we provide the proof. Lemma 4.12 Assume that p > d, and A ∈ Rd×d is symmetric, positive definite matrix. Let u be a solution of the equation − p,A (u) = 0 in a neighbourhood of infinity with lim x→∞ u(x) = ∞. Choose R > 0 and c > 0 such that vc := K [u](x) − c is positive near the origin and negative on ∂ E A (R). Then there exists C > 0 such that for any ϕ ∈ C01 (E A (R)) which equals 1 near the origin, we have  E A (R)

B[vc ] · ∇ϕ dx = C.

Proof The difference of any two such ϕ has a compact support in E A (R) \ E¯ A (0, ε) for some ε > 0. Since vc satisfies −div (B(vc )) = 0 in E A (R) \ E¯ A (0, ε), therefore it follows that  B[vc ] · ∇ϕ dx = constant = C. E A (R)

We show that the constant C is positive. For this, we choose the following test function: ⎧ ⎪ vc (x) ≤ 0, ⎨0 ϕν (x) := vc (x) 0 < vc (x) < ν, ⎪ ⎩ ν vc (x) ≥ ν. Therefore, we have   C= B[vc ] · ∇ϕ1 dx = E A (R)

β

{x∈E A (R): 0 d, and A ∈ Rd×d is a symmetric, positive definite matrix. Let vc (x) be the solution as in Lemma 4.12. Then there exists ε > 0 such that (d− p)/( p−1)

in E A (ε) \ {0}.

vc (x)

Mr =

vc |x|A−1

(4.9)

Proof For 0 < r < R, consider mr =

inf

x∈∂ E A (r )

and

sup vc (x).

x∈∂ E A (r )

Since lim vc (x) = ∞, Remark 3.15 (with v = 1) implies that the functions m r , Mr x→0

are non-decreasing when r → 0. We show that there exists constants C1 and C2 such that m r ≤ C1r (d− p)/( p−1) ≤ C2 Mr

for all 0 < r < r0 ,

for some r0 > 0. Then by applying the uniform Harnack inequality to the ( p, A)harmonic function u, the claim of the lemma will follow. Let ϕν as defined above, and note that ϕν (x) = ν near origin. Thus, by Lemma 4.12, we have  Cm r = =

B[vc ] · ∇ϕm r dx

E A (R)

C1



β

|x|A−1 |∇ϕm r |A dx ≥ C1 p

E A (R)

p

λmin

β λmax

p

m r cap p,β (Br ,R ),

where λmin and λmax denote the lowest and greatest eigenvalue of the matrix A and cap p,β (Br ,R ) is the weighted p-capacity of the ball Br in B R with respect to the measure |x|β . Then by [5, Emaple 2.2], it follows that cap p,β (Br ,R ) = C  (r ( p−d−β)/( p−1) − R ( p−d−β)/( p−1) )1− p . Since ( p − d − β)/( p − 1) = (d − p)/( p − 1), we have 1− p

Cm r

≥ C1 (r (d− p)/( p−1) − R (d− p)/( p−1) )1− p ,

which implies that   m r ≤ C1 r (d− p)/( p−1) − R (d− p)/( p−1) ≤ C1r (d− p)/( p−1) . 132

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Next we show r (d− p)/( p−1) ≤ C2 Mr , for some C2 > 0. Denote α = (d − p)/( p − 1). For 0 < r < R, consider the following test function

ψr (x) :=

⎧ ⎪ ⎨1

|x|A−1 < r ,

⎩

1 ≤ |x|A−1 ≤ R, |x|A−1 > R.

|x|α −1 −R α A ⎪ r α −R α

0

By Lemma 4.12 and using the Hölder inequality, we have 

 C=

1/ p

p β ∇ψr · B[vc ]dx ≤ |x|A−1 |∇ψr |A dx E A (R) E A (R)\E A (r )  ( p−1)/ p p β |x|A−1 |∇vc |A dx . E A (R)\E A (r )

(4.10)

Now, 

β

p

|x|A−1 |∇ψr |A dx

E A (R)\E A (r ) C

=

(r α − R α )

 (α−1) p+β+d  − R (α−1) p+β+d = r p

C , (r α − R α ) p−1

where we used (α − 1) p + β + d = α. Thus, for small r , we get 

β

|x|A−1 |∇ψr |A dx ≤ C r −α( p−1) . p

E A (R)\E A (r )

(4.11)

For the second term of (4.10), we note that vc = ψ Mr in {0 ≤ vc ≤ Mr } which is a subset of E A (R) \ E A (r ). Thus we have 

p β |x|A−1 |∇vc |A dx E A (R)\E A (r )



≤

{0≤vc ≤Mr }

β

p

|x|A−1 |∇vc |A dx ≤

 E A (R)

B[vc ] · ∇ψ Mr dx = C Mr .

(4.12)

Therefore, from (4.10), (4.11) and (4.12), we get ( p−1)/ p

C2 ≤ r α(1− p)/ p Mr which shows that r (d− p)/( p−1) = r α ≤ C2 Mr .

,  

Proof of Theorem 4.8 Let p > d. In light of Corollary 4.7, we need only to consider the case u(x) → ∞ as x → ∞. Since Lemma 4.13 implies that v(x) := K [u](x)

(d− p)/( p−1) |x|A−1 near the origin, we need to show that in fact, v(x) := K [u](x) ∼ Reprinted from the journal

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R. Kr. Giri ,Y. Pinchover (d− p)/( p−1)

( p−d)/( p−1)

|x|A−1 as x → 0. Then in light of Lemma 4.10, u(x) ∼ |x|A−1 as x → ∞. We follow the proof of [2, Theorem 2.3]. For 0 < σ < 1, define wσ (x) := v(σ x)/σ α where α = (d − p)/( p − 1). Since vc |x|αA−1 in E A (ε) \ {0}, it follows that wσ (x) |x|αA−1 in E A (ε/σ ) \ {0} for some ε > 0 and also the family {wσ }0 0.

Hence, m|x|αA−1 ≤ w(x) ≤ M|x|αA−1

∀R > 0.

Note that |x|αA−1 is a positive solution of − div (B(u)) = 0 in Rd \ {0} and the function |x|αA−1 does not have any critical point. Hence, by the strong comparison principle (see, [2, Theorem 3.2]) which is valid also for the ( p, A)-operator, we obtain   m|x|αA−1 = w(x) = M|x|αA−1 , and hence, m = M. Proof of Theorem 4.1 The proof follows directly from Theorems 4.4 and 4.8. 134

 

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Positive Liouville theorem and asymptotic behaviour for…

5 Weak Fuchsian singularity and positive Liouville theorems In this section we introduce the notion of weak Fuchsian singularity, and prove Conjecture 3.10 for Q which has weak Fuchsian singularity at ζ (see, Theorem 5.4). Definition 5.1 Let A and V satisfy Assumptions 2.3. Assume that Q has an isolated ˆ where ζ = 0 or ζ = ∞. The operator Q = Q p,A,V is Fuchsian singularity ζ ∈ ∂ , ( j) said to have a weak Fuchsian singularity at ζ if there exist m sequences {Rn }∞ n=1 ⊂ ( j) j (1) ( j) ( j) R+ , 1 ≤ j ≤ m, satisfying Rn → ζ , where ζ = ζ , and ζ = 0 or ζ = ∞ for 2 ≤ j ≤ m, such that (m)

D{Rn

}

(1)

◦ · · · ◦ D{Rn } (Q)(w) = − p,A (w)

on Y ,

(5.1)

where A ∈ Rd×d is a symmetric, positive definite matrix, and Y = lim /Rn(1) . n→∞

Remark 5.2 Example 2.1 in [2] demonstrates that m in (5.1) might be greater than 1. Moreover, although in this example V ∈ / M q ( p; B1 \ {0}), the corresponding operator has a weak Fuchsian singularity at ζ = 0. The next example shows that if ζ = 0 and V ∈ M q ( p; ) for some punctured neighborhood  of the origin, and A is continuous at 0, then Q has weak Fuchsian singularity at ζ = 0. d×d ) is continuous at the isolated singular Example 5.3 Assume that A ∈ L ∞ loc (; R q ˆ Further point ζ = 0. Let V ∈ Mloc ( p; ) have a Fuchsian singularity at 0 ∈ ∂ . suppose that V ∈ M q ( p; B1 ∩ ). Then for any smooth function ϕ having compact support in Br \{0} we have

   

  R V (Rx)ϕ(x)dx  /R   ≤ R p−d |V (x)||ϕ(x/R)|dx ≤ R p−d ϕ ∞ p



∩B Rr

|V (x)|dx.

(5.2)

Take R > 0 small enough such that  ∩ B Rr ⊂  ∩ B1 . Then for p < d, (5.2) implies    

/R

   R p V (Rx)ϕ(x)dx  ≤ R p−d (Rr )d/q ϕ ∞

1  (Rr )d/q

 ∩B Rr

|V (x)|dx



≤ ϕ ∞r d/q V M q ( p;∩B1 ) R p−d/q −→ 0, R→0

while for p > d    

  R V (Rx)ϕ(x)dx  ≤ ϕ ∞ V L 1 (∩B1 ) R p−d −→ 0. R→0 p

/R

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R. Kr. Giri ,Y. Pinchover

Similarly, for p = d it can be seen that    

/R

  R p V (Rx)ϕ(x)dx  ≤ ϕ ∞ V M q (d;∩B1 )

1

−→ 0.

 logq/d (1/Rr ) R→0

Therefore, the operator Q p,A,V has a weak Fuchsian singularity at 0. Theorem 5.4 (Liouville theorem) Let A and V satisfy Assumptions 2.3. Suppose that ˆ is an isolated singular point. Assume that the operator Q = Q p,A,V has a ζ ∈ ∂ weak Fuchsian singularity at ζ . Then ζ is a regular point of Eq. (2.1). In other words, if u and v are two positive solutions of the equation Q p,A,V (w) = 0 in a punctured neighbourhood of ζ , then u(x) exists in the wide sense. x→ζ v(x) (ii) the equation Q p,A,V (w) = 0 admits a unique positive solution in  of minimal ˆ \ {ζ }. growth in a neighbourhood of ∂  (i)

lim

Proof By Proposition 3.17, we have (i) ⇒ (ii). Thus, we only need to show that lim u(x) v(x) exists in the wide sense. Since the operator Q has a weak Fuchsian singularity x→ζ

at ζ , we have (m)

D{Rn

}

(1)

◦ · · · ◦ D{Rn } (Q)(w) = − p,A (w) = 0

in Rd \ {0},

(5.3)

where A ∈ Rd×d is a symmetric, positive definite matrix. Recall that by Theorem 4.1 either 0 or ∞ is a regular point of − p,A . Therefore, Proposition 3.18 and a reverse induction argument implies that ζ is a regular point of the equation Q p,A,V (w) = 0.  

6 Positive Liouville theorem in the elliptically symmetric case This section is devoted to the proof of Conjecture 3.10 in the elliptically symmetric case. Definition 6.1 Let A ∈ Rd×d be a symmetric, positive definite matrix. We say that f :  → R is elliptically symmetric with respect to A if f (x) = f˜(|x|A−1 ) for all x ∈ , where f˜ : R+ → R. In the sequel, with some abuse of notation, we omit the distinction between f and f˜. Throughout the present section we fix A ∈ Rd×d and assume that the potential V ∈ q Mloc ( p; ) is elliptically symmetric with respect to A i.e., V (x) = V (|x|A−1 ). p−2 Denote r = |x|A−1 , and let us calculate  p,A ( f (r )) = div(|∇ f (r )|A A∇ f (r )). Using (4.2), we obtain ∇ f (r ) = f  (r )

A−1 x r

and |∇ f (r )|A = 136

| f  (r )| −1 |A x|A = | f  (r )|. r Reprinted from the journal

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Consequently, x p−2 η := |∇ f (r )|A A∇ f (r ) = | f  (r )| p−2 f  (r ) , and r ∂ηi | f  (r )| p−2 f  (r ) = ∂ xi r   −1 | f  (r )| p−2 f  (r ) | f  (r )| p−2 f  (r ) xi (A x)i − . + ( p − 1) + r r2 r Therefore, we get  p,A ( f (r )) =

d  ∂ηi

∂ xi i=1   d −1  f (r ) , where r = |x|A−1 . = | f  (r )| p−2 ( p − 1) f  (r ) + r

(6.1)

Lemma 6.2 Let A ∈ Rd×d be a symmetric, positive definite matrix. Assume that the domain  and the potential V are elliptically symmetric with respect to A and the equation Q p,A,V (u) = 0 possess a positive solution. Further, suppose that the operator Q p,A,V has a Fuchsian isolated singularity at ζ ∈ {0, ∞}. Then for any u ∈ Gζ , there exists an elliptically symmetric (with respect to A) solution u˜ ∈ Gζ such that u u. ˜ Proof We consider the case ζ = 0, the case when ζ = ∞, can be shown similarly. Fix R > 0 such that u is defined in the punctured ellipsoid E A (2R) \ {0}. Then for 0 < ρ < R, consider the following Dirichlet problem ⎧ ⎪ ⎨ Q p,A,V (w) = 0 in E A (R) \ E¯ A (ρ), x ∈ ∂ E A (R) w(x) = m R ⎪ ⎩ x ∈ ∂ E A (ρ), w(x) = m ρ

(6.2)

where m r = inf x∈∂ E A (r )u(x). By Lemma 2.5, there exists a unique solution u ρ,R to the Dirichlet problem (6.2). Moreover, from the unique solvability of the one-dimensional Dirichlet problem it follows that u ρ,R is elliptically symmetric with respect to A. Moreover, by the uniform Harnack inequality (Theorem 3.13) and the WCP we have u ρ,R ≤ u ≤ Cu ρ,R in E A (R) \ E A (ρ), where C > 0 is independent of ρ. Applying the Harnack converging principle, it follows that there exists a sequence ρn → 0 such that u ρn ,R → u˜ locally uniformly in E A (R)\{0}, where u˜ is an elliptically   symmetric positive solution of the equation Q p,A,V (w) = 0 in E A (R) \ {0}. Reprinted from the journal

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Theorem 6.3 Let A ∈ Rd×d be a symmetric, positive definite matrix. Assume that the domain  and the potential V are elliptically symmetric with respect to A and the corresponding equation (2.1) possess a positive solution. Further, suppose that the operator Q p,A,V has a Fuchsian isolated singularity at ζ ∈ {0, ∞}. Then (i) ζ is a regular point of (2.1). (ii) the equation Q p,A,V (w) = 0 possess a unique positive solution in  of minimal ˆ \ {ζ }. growth in a neighbourhood of ∂  Proof (i) Assume first that u, v ∈ Gζ , where u is elliptically symmetric with respect to A. Since the operator Q p,A,V has a Fuchsian isolated singularity at ζ , hence Lemma 3.14, the proof of Proposition 3.18, and the uniform Harnack inequality Theorem 3.13, imply that either lim

u(x)

x→ζ v(x) x∈

exists, and equal either to 0 or ∞,

or else, u v in some punctured neighbourhood  ⊂  of ζ . For a sequence {Rn } which converges to ζ , define u n (x) and vn (x) as in the proof of Proposition 3.18 (see, (3.14)). Then, u n and vn are positive solutions of (3.15). Following the arguments as in Proposition 3.18, it follows that up to a subsequence lim u n (x) = u ∞ (x), lim vn (x) = v∞ (x),

n→∞

n→∞

locally uniformly in Rd \ {0}, and u ∞ , v∞ are positive solutions of the limiting dilated equation − p,A (w) + V|w| p−2 w = 0 in Rd \ {0}. Note that the potential V and the solution u ∞ are elliptically symmetric with respect to A. Moreover, as in Proposition 3.18, for any fixed R > 0, we have u ∞ (x) u ∞ (x) = M, inf = m, x∈S R v∞ (x) x∈S R v∞ (x) sup

where M = limr →ζ Mr and m = limr →ζ m r and m r , Mr are defined as in Lemma 3.14. Assume that the potential V is nonzero, otherwise it has a weak Fuchsian singularity at ζ and the theorem follows from Theorem 5.4. Let Su ∞ be the set of critical points of u ∞ . Then it is closed and elliptically symmetric. Now if ζ is an interior point of Sˆu ∞ then |∇u ∞ | = 0 in some punctured neighbourhood  of ζ . This implies that u ∞ is constant near ζ which contradicts our assumption that V = 0 near ζ . Therefore, there exists an annular set A˜ = E A (R) \ E A (r ) close to ζ such that Su ∞ ∩ A˜ = ∅. Hence by the strong comparison principle (see [2, Theorem 2]), which is also valid for Q p,A,V , we obtain ˜ So, m = M, and the theorem follows. mv∞ = u ∞ = Mv∞ in A. 138

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Assume now that u, v ∈ Gζ . Then by Lemma 6.2, there exists a elliptically sym˜ By the proof before if follows that u ∼ u˜ and metric solution u˜ ∈ Gζ such that u u. the limit lim

v(x)

x→ζ u(x) ˜ x∈

exists in the wide sense, and lim

u(x)

x→ζ u(x) ˜ x∈

= C > 0,

which shows that u(x) exists in the wide sense. x→ζ v(x) lim

x∈

 

(ii) Follows from Proposition 3.17.

Acknowledgements The authors acknowledge the support of the Israel Science Foundation (Grant 637/19) founded by the Israel Academy of Sciences and Humanities.

Compliance with ethical standards Conflict of interest The authors declare that they have no conflict of interest.

References 1. Agmon, S.: On positivity and decay of solutions of second order elliptic equations on Riemannian manifolds. In: Methods of Functional Analysis and Theory of Elliptic Equations (Liguori), pp. 19–52 (1982) 2. Frass, M., Pinchover, Y.: Positive Liouville theorems and asymptotic behavior for p-Laplacian type elliptic equations with a Fuchsian potential. Confluentes Mathematici 3, 291–323 (2011) 3. Fraas, M., Pinchover, Y.: Isolated singularities of positive solutions of p-Laplacian type equations in Rd . J. Differ. Equ. 254, 1097–1119 (2013) 4. Guedda, M., Véron, L.: Local and global properties of solutions of quasilinear elliptic equations. J. Differ. Equ. 76, 159–189 (1988) 5. Heinonen, J., Kilpeläinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations, Unabridged republication of the 1993 original. Dover Publications Inc, Mineola (2006) 6. Kichenassamy, S., Véron, L.: Singular solutions of the p-Laplace equation. Math. Ann. 275, 599–615 (1986) 7. Kichenassamy, S., Véron, L.: Erratum: “Singular solutions of the p-Laplace equation”. Math. Ann. 277, 352 (1987) 8. Malý, J., Ziemer, W.P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations, Mathematical Survey and Monographs 51. American Mathematical Society, Providence (1997) 9. Maz’ya, V.G.: The continuity at a boundary point of the solutions of quasi-linear elliptic equations (Russian). Vestnik Leningrad. Univ. 25(13), 42–55 (1970) 10. Maz’ya, V.: Seventy five (thousand) unsolved problems in analysis and partial differential equations. Integral Equ. Oper. Theory 90 , Paper No. 25 (2018) 11. Moreira, D.R., Teixeira, E.V.: On the behavior of weak convergence under nonlinearities and applications. Proc. Am. Math. Soc. 133, 1647–1656 (2004) 12. Padberg, M.: Linear Optimization and Extensions, Second Revised and Expanded edition, Algorithms and Combinatorics, 12. Springer, Berlin (1999) 13. Pinchover, Y.: On positive Liouville theorems and asymptotic behavior of solutions of Fuchsian type elliptic operators. Ann. Inst. Henri Poincaré Anal. Non Linéaire 11, 313–341 (1994)

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R. Kr. Giri ,Y. Pinchover 14. Pinchover, Y., Psaradakis, G.: On positive solutions of the ( p, A)-Laplacian with potential in Morrey space. Anal. PDE 9, 1357–1358 (2016) 15. Pinchover, Y., Regev, N.: Criticality theory of half-linear equations with the ( p, A)-Laplacian. Nonlinear Anal. 119, 295–314 (2015) 16. Pinchover, Y., Tintarev, K.: Ground state alternative for p-Laplacian with potential term. Calc. Var. Partial Differ. Equ. 28, 179–201 (2007) 17. Pucci, P., Serrin, J.: The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications 73. Birkhäuser, Basel (2007) 18. Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta Math. 111, 247–302 (1964) 19. Serrin, J.: Isolated singularities of solutions of quasi-linear equations. Acta Math. 113, 219–240 (1965) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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Analysis and Mathematical Physics (2020) 10:78 https://doi.org/10.1007/s13324-020-00420-0

Space quasiconformal composition operators with applications to Neumann eigenvalues Vladimir Gol’dshtein1 · Ritva Hurri-Syrjänen2 · Valerii Pchelintsev3,4 · Alexander Ukhlov1 Received: 26 August 2020 / Revised: 21 October 2020 / Accepted: 23 October 2020 / Published online: 9 November 2020 © Springer Nature Switzerland AG 2020

Abstract In this article we obtain estimates of Neumann eigenvalues of p-Laplace operators in a large class of space domains satisfying quasihyperbolic boundary conditions. The suggested method is based on composition operators generated by quasiconformal mappings and their applications to Sobolev–Poincaré-inequalities. By using a sharp version of the reverse Hölder inequality we refine our estimates for quasi-balls, that is, images of balls under quasiconformal mappings of the whole space. Keywords Elliptic equations · Sobolev spaces · Quasiconformal mappings Mathematics Subject Classification 35P15 · 46E35 · 30C65

B

Alexander Ukhlov [email protected] Vladimir Gol’dshtein [email protected] Ritva Hurri-Syrjänen [email protected] Valerii Pchelintsev [email protected]

1

Department of Mathematics, Ben-Gurion University of the Negev, P.O. Box 653, Beer Sheva 8410501, Israel

2

Department of Mathematics and Statistics, University of Helsinki, Gustaf Hällströmin katu 2 b, 00014 Helsinki, Finland

3

Division for Mathematics and Computer Sciences, Tomsk Polytechnic University, Lenin Ave. 30, Tomsk, Russia 634050

4

Department of Mathematical Analysis and Theory of Functions, Tomsk State University, Lenin Ave. 36, Tomsk, Russia 634050

Chapter 6 was originally published as Gol’dshtein, V., Hurri-Syrjänen, R., Pchelintsev, V. & Ukhlov, A. Analysis and Mathematical Physics (2020) 10:78. https://doi.org/10.1007/s13324-020-00420-0. Reprinted from the journal

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1 Introduction The article is devoted to applications of the space quasiconformal mappings theory to the spectral theory of elliptic operators. Applications are based on the geometric theory of composition operators on Sobolev spaces [19,33,35] in the special case of operators generated by quasiconformal mappings. Composition operators on Sobolev spaces permit us to give estimates of norms for embedding operators of Sobolev spaces into Lebesgue spaces in a large class of space domains that includes domains with Hölder singularities [21,22]. Quasiconformal mappings allow us to describe the important subclass of these embedding domains in the terms of quasihyperbolic geometry. It permits us to obtain estimates of Neumann eigenvalues of the p-Laplace operator, p > n, in domains with quasihyperbolic boundary conditions. These estimates are refined for K -quasi-balls, that is, images of the ball B ⊂ Rn under K -quasiconformal mappings ϕ : Rn → Rn with the help of a sharp (with estimates of constants) reverse Hölder inequality. We prove the estimates of constants in the reverse Hölder inequality on the base of the quasiconformal mapping theory [4] and the non-linear potential theory [23,30]. Namely we prove that if a domain Ω ⊂ Rn , n ≥ 3, satisfying the γ -quasihyperbolic boundary condition, is a K -quasi-ball, then for p > n the first non-trivial Neumann eigenvalue μ p (Ω) satisfies to the inequality: μ p (Ω) ≥

M p (K , γ ) , p R∗

where R∗ is a radius of a ball Ω ∗ of the same measure as Ω and M p (K , γ ) depends only on p, γ and the quasiconformity coefficient K of Ω. The exact value of M p (K , γ ) is given in Theorem 12. Estimates of the first non-trivial Neumann eigenvalue of the p-Laplace operator, p > 2, are known for convex domains Ω ⊂ Rn [7]:  μ p (Ω) ≥

πp d(Ω)

p , 1

where d(Ω) is a diameter of a convex domain Ω, π p = 2π ( p − 1) p /( p sin(π/ p)). Unfortunately in non-convex domains μ p (Ω) can not be characterized in the terms of Euclidean diameters. This can be seen by considering a domain consisting of two identical squares connected by a thin corridor [5]. The method of the composition operators on Sobolev spaces in connection with embedding theorems in convex domains [12,18] allows us to obtain estimates of Neumann eigenvalues of the p-Laplace operator in non-convex domains including some domains with Hölder singularities and some fractal type domains [20,21]. In the case of composition operators generated by quasiconformal mappings the main estimates of norms of embedding operators from Sobolev spaces with first derivatives into Lebesgue spaces were reformulated in the terms of integrals of quasiconformal derivatives [13,14,16]. This type of global integrability of quasiconformal 142

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derivatives depends on the quasiconformal geometry of domains [8]. One of the possible geometric reinterpretation of the quasiconformal geometry is the growth condition for quasihyperbolic metric [2,26]. Recall that a domain Ω satisfies the γ -quasihyperbolic boundary condition with some γ > 0, if the growth condition on the quasihyperbolic metric kΩ (x0 , x) ≤

1 dist(x0 , ∂Ω) log + C0 γ dist(x, ∂Ω)

is satisfied for all x ∈ Ω, where x0 ∈ Ω is a fixed base point and C0 = C0 (x0 ) < ∞, [9,24,25,29]. In [2] it was proved that Jacobians Jϕ of quasiconformal mappings ϕ : B → Ω belong to L β (B) for some β > 1 if and only if Ω satisfies a γ -quasihyperbolic boundary condition with some γ which depends on K , n and β. Hence the (quasi)conformal mapping theory allows to obtain spectral estimates in domains with quasihyperbolic boundary conditions. Because we need the exact value of the integrability exponent β for quasiconformal Jacobians, we consider an equivalent class of β-quasiconformal regular domains [17], namely the class of domains 

 Ω ⊂ Rn : Ω = ϕ(B) with Jϕ ∈ L β (B) .

An important subclass of β-quasiconformal regular domains are quasi-balls, because Jacobians of quasiconformal mappings are A p -weights [32] and they satisfy the reverse Hölder inequality [4]. Methods of the harmonic analysis allow us to refine estimates of Neumann eigenvalues in the case of quasi-balls. The main technical difficulties are calculation (estimation) of exact constants in the reverse Hölder inequality. We solve this problem using capacitary (moduli) estimates. These estimates permit us to obtain integral estimates of quasiconformal derivatives (Jacobians) in the unit ball B. Hence we can refine estimates of the Neumann eigenvalues for quasi-balls. Note that quasi-balls include some fractal type domains. In the two-dimensional case R2 this approach is more accurate [15,16] because exact exponents of local integrability of planar quasiconformal Jacobians are known [3,11].

2 Composition operators on Sobolev spaces 2.1 Sobolev spaces Let Ω be an open subset of Rn , n ≥ 2. The Sobolev space W p1 (Ω), 1 ≤ p ≤ ∞ is defined as a Banach space of locally integrable weakly differentiable functions f : Ω → R equipped with the following norm: f | W p1 (Ω) = f | L p (Ω) + ∇ f | L p (Ω) , Reprinted from the journal

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where ∇ f is the weak gradient of the function f . The homogeneous seminormed Sobolev space L 1p (Ω), 1 ≤ p ≤ ∞ is equipped with the seminorm: f | L 1p (Ω) = ∇ f | L p (Ω) . We consider the Sobolev spaces as Banach spaces of equivalence classes of functions up to a set of p-capacity zero [30]. Recall that a homeomorphism ϕ : Ω → Ω  is called a K -quasiconformal mapping 1 (Ω) and there exists a constant 1 ≤ K < ∞ such that if ϕ ∈ Wn,loc |Dϕ(x)|n ≤ K |J (x, ϕ)| for almost all x ∈ Ω. The quasiconformal mappings are mappings of finite distortion and possess the Luzin N -property, that is, an image of a set of measure zero has measure zero. Note, that if ϕ : Ω → Ω  is a quasiconformal mapping, then the volume derivative Jϕ (x) := lim

r →0+

|ϕ(B(x, r ))| = |J (x, ϕ)| for almost all x ∈ Ω. |B(x, r )|

Let Ω and Ω  be domains in Rn , that is, open and connected sets. Then Ω and Ω  are called K -quasiconformal equivalent domains if there exists a K -quasiconformal homeomorphism ϕ : Ω → Ω  . Note that in R2 any two simply connected domains are quasiconformal equivalent domains [1]. In the following theorems we obtain an estimate of the norm of the composition operator on Sobolev spaces in quasiconformal equivalent domains with finite measure. The case of planar conformal mappings was considered in [16]. Theorem 1 Let Ω, Ω  ⊂ Rn , n ≥ 2, be K -quasiconformal equivalent domains with finite measure. Then a K -quasiconformal mapping ϕ : Ω → Ω  generates a bounded composition operator ϕ ∗ : L 1p (Ω  ) → L q1 (Ω) 1

for any p ∈ (n, +∞) and q ∈ [1, n] with K p,q (Ω) = K n |Ω  |

p−n np

|Ω|

n−q nq

.

Proof By using the quasiconformal inequality |Dϕ(x)|n ≤ K |J (x, ϕ)| for almost all x ∈ Ω we obtain ⎛ ⎞ p−q pq  q   p p−q |Dϕ(x)| K p,q (Ω) = ⎝ dx⎠ |J (x, ϕ)| Ω

⎞ p−q ⎛ pq  ( p−n)q 1 p p−q ⎠ ⎝ ≤K |Dϕ(x)| dx . Ω

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Note that if q ≤ n then the quality ( p − n)q/( p − q) ≤ n. Hence applying the Hölder inequality to the last integral we have p−q ⎡⎛ ⎛ ⎞ p−q ⎞ ( p−n)q ⎛ ⎞ (n−q) p ⎤ pq pq n( p−q) n( p−q)    ( p−n)q ⎥ ⎢ ⎝ |Dϕ(x)| p−q d x ⎠ ⎝ dx⎠ ≤ ⎣⎝ |Dϕ(x)|n d x ⎠ ⎦

Ω

Ω

≤K

p−n np

Ω p−q ⎡⎛ ⎞ ( p−n)q ⎛ ⎞ (n−q) p ⎤ pq n( p−q) n( p−q)   ⎢⎝ ⎥ ⎝ dx⎠ |J (x, ϕ)| d x ⎠ . ⎣ ⎦

Ω

Ω

By the condition of the theorem, Ω and Ω  are Euclidean domains with finite measure and therefore 1

K p,q (Ω) ≤ K p K

p−n np

|Ω  |

p−n np

|Ω|

n−q nq

1

= K n |Ω  |

p−n np

|Ω|

n−q nq

< ∞.

Hence, by [33,35] we obtain that a composition operator ϕ ∗ : L 1p (Ω  ) → L q1 (Ω) is bounded for any p ∈ (n, +∞) and q ∈ [1, n].



Let Ω and Ω  be domains in Rn , n ≥ 2. Then a domain Ω  is called a K -quasiconformal β-regular domain about a domain Ω if there exists a K -quasiconformal mapping ϕ : Ω → Ω  such that Jϕ | L β (Ω) < ∞ for some β > 1, where Jϕ (x) = J (x, ϕ) is a Jacobian of a K -quasiconformal mapping ϕ : Ω → Ω  . The domain Ω  ⊂ Rn is called a quasiconformal regular domain if it is a K quasiconformal β-regular domain for some β > 1. Note that the class of quasiconformal regular domains includes the class of Gehring domains [2,26] and can be described in terms of quasihyperbolic geometry [9,24,29]. Now we establish a connection between quasiconformal β-regular domains and the quasiconformal composition operators on Sobolev spaces. Theorem 2 Let Ω, Ω  ⊂ Rn , n ≥ 2, be domains. Then Ω  is a K -quasiconformal β-regular domain about a domain Ω, 1 < β < ∞, if and only if a K -quasiconformal mapping ϕ : Ω → Ω  generates a bounded composition operator ϕ ∗ : L 1p (Ω  ) → L q1 (Ω) for any p ∈ (n, +∞) and q = pβ/( p + β − n). Reprinted from the journal

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V. Gol’dshtein et al. 1 (Ω) and Jacobian Proof Because ϕ is a quasiconformal mapping, then ϕ ∈ Wn,loc J (x, ϕ) = 0 for almost all x ∈ Ω. Hence the p-dilatation

K p (x) =

|Dϕ(x)| p |J (x, ϕ)|

is well defined for almost all x ∈ Ω and so ϕ is a mapping of finite distortion. Let Ω  be a K -quasiconformal β-regular domain about a domain Ω. Then there exists K -quasiconformal mapping ϕ : Ω → Ω  such that 

|J (x, ϕ)|β d x < ∞ for some β > 1.

Ω

Taking into account the quasiconformal inequality |Dϕ(x)|n ≤ K |J (x, ϕ)| for almost all x ∈ Ω we obtain pq p−q

K p,q

 

 q  q   p−q |Dϕ(x)| p p−q |Dϕ(x)|n p−n |Dϕ(x)| (Ω) = dx = dx |J (x, ϕ)| |J (x, ϕ)| Ω Ω   q q   q p−n p−q d x = K p−q ≤ K p−q |Dϕ(x)| |Dϕ(x)|β d x < ∞, Ω

Ω

for β = ( p −n)q/( p −q). Hence by [33,35] we have a bounded composition operator ϕ ∗ : L 1p (Ω  ) → L q1 (Ω) for any p ∈ (n, +∞) and q = pβ/( p + β − n). Let us check that q < p. Because p > n we have that p + β − n > β > 1 and so β/( p + β − n) < 1. Hence we obtain q < p. Assume that the composition operator ϕ ∗ : L 1p (Ω  ) → L q1 (Ω), q < p, is bounded for any p ∈ (n, +∞) and q = pβ/( p + β − n). Then, given the Hadamard inequality: |J (x, ϕ)| ≤ |Dϕ(x)|n for almost all x ∈ Ω, and by [33,35] we have 

β



|Dϕ(x)| d x = Ω

|Dϕ(x)|

( p−n)q p−q

  dx ≤

Ω

Ω

|Dϕ(x)| p |J (x, ϕ)|



q p−q

d x < +∞.



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Remark 1 In the case of bounded domains Ω, Ω  ⊂ Rn , n ≥ 2, Theorem 2 is correct for any p ∈ (n, +∞) and any q ≤ pβ/( p + β − n). If q > n − 1, then by the duality composition theorem [33], the inverse mapping ϕ −1 induces a bounded composition operator from L q1  (Ω) to L 1p (Ω  ), where p  = p/( p − (n − 1)) and q  = q/(q − (n − 1)).

3 Sobolev–Poincaré inequalities 3.1 Two-weight Sobolev–Poincaré inequalities Let Ω ⊂ Rn , n ≥ 2, be a domain and let h : Ω → R be a real valued locally integrable function such that h(x) > 0 a.e. in Ω. We consider the weighted Lebesgue space L p (Ω, h), 1 ≤ p < ∞, as the space of measurable functions f : Ω → R with the finite norm ⎛ ⎞1 p  f | L p (Ω, h) := ⎝ | f (x)| p h(x) d x ⎠ < ∞. Ω

It is a Banach space for the norm f | L p (Ω, h) . On the basis of Theorem 1 we prove the existence of two-weight Sobolev–Poincaré inequalities in quasiconformal equivalent bounded space domains. Recall that a bounded domain Ω ⊂ Rn is called a (r , q)-Sobolev–Poincaré domain, 1 ≤ r , q ≤ ∞, if for any function f ∈ L q1 (Ω), the (r , q)-Sobolev–Poincaré inequality inf f − c | L r (Ω) ≤ Br ,q (Ω) ∇ f | L q (Ω)

c∈R

holds. Note that bounded Lipschitz domains Ω ⊂ Rn are (r , q)-Sobolev–Poincaré domains, for 1 ≤ r ≤ nq/(n − q), if 1 ≤ q < n, and for any r ≥ 1, if q ≥ n (see, for example, [37, Theorem 2.5.1]). Theorem 3 Let Ω, Ω  ⊂ Rn , n ≥ 2, be K -quasiconformal equivalent bounded domains and let h(y) = |J (y, ϕ −1 )| be the quasiconformal weight defined by a K quasiconformal mapping ϕ : Ω → Ω  . Suppose that Ω be a (r , q)-Sobolev–Poincaré domain, then for any function f ∈ W p1 (Ω  ), p > n, the inequality ⎞1 ⎛ ⎞1 ⎛ r p   r  p inf ⎝ | f (y) − c| h(y) dy ⎠ ≤ Br , p (Ω , h) ⎝ |∇ f (y)| dy ⎠

c∈R

Ω

Ω

holds for any 1 ≤ r ≤ nq/(n − q) with the constant Br , p (Ω, h) ≤ inf

q∈[1,n]

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147

n−q nq



1

K n |Ω  |

p−n np

.

V. Gol’dshtein et al.

Here Br ,q (Ω) is the best constant in the (unweighted) (r , q)-Sobolev–Poincaré inequality in the domain Ω. Proof By the conditions of the theorem there exists a K -quasiconformal mapping ϕ : Ω → Ω  . Denote by h(y) = |J (y, ϕ −1 )| the quasiconformal weight in Ω  . Let f ∈ L 1p (Ω  ) be a smooth function. Then the composition g = f ◦ ϕ −1 is well defined almost everywhere in Ω and belongs to the Sobolev space L q1 (Ω) [34]. Hence, because Ω is the (r , q)-Sobolev–Poincaré domain we have that g = f ◦ϕ −1 ∈ Wq1 (Ω) and the unweighted Poincaré–Sobolev inequality inf || f ◦ ϕ −1 − c | L r (Ω)|| ≤ Br ,q (Ω)||∇( f ◦ ϕ −1 ) | L q (Ω)||

c∈R

(1)

holds for any 1 ≤ r ≤ nq/(n − q). Taking into account the change of variable formula for quasiconformal mappings [34], the Poincaré–Sobolev inequality (1) and Theorem 1, we obtain for ⎛ ⎞1 r  inf ⎝ | f (y) − c|r h(y)dy ⎠

c∈R

Ω

⎞1 ⎛ r  r −1 ⎠ ⎝ = inf | f (y) − c| |J (y, ϕ )|dy c∈R

Ω

⎛ ⎞1 ⎛ ⎞1 r q   r q ⎝ ⎠ ⎝ ⎠ = inf |g(x) − c| d x ≤ Br ,q (Ω) |∇g(x)| d x c∈R

Ω

⎛ 1 n

≤ Br ,q (Ω)K |Ω|

n−q nq

|Ω  |

p−n np

⎝

Ω



⎞1

p

|∇ f (y)| p dy ⎠ .

Ω

Approximating an arbitrary function f ∈ W p1 (Ω  ) by smooth functions we have ⎞1 ⎛ ⎞1 ⎛ r p   inf ⎝ | f (y) − c|r h(y)dy ⎠ ≤ Br , p (Ω  , h) ⎝ |∇ f (y)| p dy ⎠

c∈R

Ω

Ω

with the constant Br , p (Ω, h) ≤ inf

q∈[1,n]



Br ,q (Ω)|Ω|

n−q nq



1

K n |Ω  |

p−n np

.



148

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Space quasiconformal composition operators with…

The property of the K -quasiconformal β-regularity implies the integrability of a Jacobian of quasiconformal mappings and therefore for any K -quasiconformal βregular domain we have the embedding of weighted Lebesgue spaces L r (Ω, h) into non-weight Lebesgue spaces L s (Ω) for s = (β − 1)r /β [17]: Lemma 1 Let Ω  be a K -quasiconformal β-regular domain about a domain Ω. Then for any function f ∈ L r (Ω, h), β/(β − 1) ≤ r < ∞, the inequality ⎛ || f | L s (Ω  )|| ≤ ⎝



⎞ 1 ·1 β s β   J (x, ϕ) d x ⎠ || f | L r (Ω  , h)||

Ω

holds for s = (β − 1)r /β. According to Theorem 3 and Lemma 1 we obtain an upper estimate of the Poincaré constant in quasiconformal regular domains. Theorem 4 Let Ω  be a K -quasiconformal β-regular domain about a (r , q)-Sobolev– Poincaré domain Ω. Then for any function f ∈ W p1 (Ω  ), p > n, the Poincaré–Sobolev inequality ⎛ inf ⎝

c∈R



⎞1

⎛

s

| f (y) − c| h(y)dy ⎠ ≤ Bs, p (Ω  ) ⎝



s

Ω

⎞1

p

|∇ f (y)| dy ⎠ p

Ω

holds for any 1 ≤ s ≤ nq/(n − q) · (β − 1)/β with the constant ⎛ ⎞ 1 ·1 β s    β    ⎝ ⎠ J (x, ϕ) d x Bs, p (Ω ) ≤ Br , p (Ω, h) Ω

≤

inf∗

q∈(q ,n]



Br ,q (Ω)|Ω|

n−q nq



1

K n |Ω  |

p−n np

1

· ||Jϕ | L β (Ω)|| s ,

where q ∗ = βns/(βs + β(n − 1)) Proof Let f ∈ W p1 (Ω  ), p > n. Then by Theorem 3 and Lemma 1 we obtain ⎞1 ⎛ s  s inf ⎝ | f (y) − c| dy ⎠

c∈R

Ω

⎛ ⎞ 1 ·1 ⎞1 ⎛ β s r   β  r   ⎝ ⎠ ⎠ ⎝ J (x, ϕ) d x ≤ inf | f (y) − c| h(y) dy Ω

Reprinted from the journal

c∈R

149

Ω

V. Gol’dshtein et al.

⎛ ≤ Br , p (Ω  , h) ⎝



⎞ 1 ·1 ⎛ ⎞1 β s p  β   J (x, ϕ) d x ⎠ ⎝ |∇ f (y)| p dy ⎠ Ω

Ω

for 1 ≤ s ≤ nq/(n − q) · (β − 1)/β. Since by Lemma 1 s = β−1 β r and by Theorem 3 r ≥ 1, then s ≥ 1 and the theorem proved.

By the generalized version of the Rellich–Kondrachov compactness theorem (see, for example, [30]) and the (r , p)–Sobolev–Poincaré inequality for r > p, it follows that the embedding operator i : W p1 (Ω) → L p (Ω) is compact in K -quasiconformal β-regular domains Ω ⊂ Rn , n ≥ 2. Note that sufficient conditions for validity of the Rellich–Kondrachov theorem in non-smooth domains have been given by using a general quasihyperbolic boundary condition in [6]. In particular, for domains Ω ⊂ Rn , n ≥ 2, satisfying a quasihyperbolic boundary condition it is proved that there exists p0 = p0 (Ω) < n such that the embedding operator i : W p1 (Ω) → L p (Ω) is compact for every p > p0 . So, by the Min–Max Principle the first non-trivial Neumann eigenvalue μ p (Ω) can be characterized [7] as ⎧ ⎫ p ⎪ ⎪  ⎨ |∇u(x)| d x ⎬ Ω 1 p−2 μ p (Ω) = min  : u ∈ W (Ω) \ {0}, |u| u d x = 0 . p p ⎪ ⎪ ⎩ |u(x)| d x ⎭ Ω

Ω

In the case s = p Theorem 4 and the Min–Max Principle implies the lower estimates of the first non-trivial eigenvalue of the degenerate p-Laplace Neumann operator, p > n, in K -quasiconformal β-regular domains Ω  ⊂ Rn , n ≥ 2. Theorem 5 Let Ω  be a K -quasiconformal β-regular domain about a (r , q)-Sobolev– Poincaré domain Ω, r = pβ/(β − 1), p > n. Then the following inequality holds   p(n−q) p p−n 1 p nq B K n |Ω  | n · ||Jϕ | L β (Ω)||, ≤ inf (Ω)|Ω| r ,q  ∗ μ p (Ω ) q∈(q ,n] where q ∗ = βnp/(β p + n(β − 1)). In the case of K -quasiconformal ∞-regular domains, in the similar way we obtain the following assertion: 150

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Theorem 6 Let Ω  be a K -quasiconformal ∞-regular domain about a ( p, q)Sobolev–Poincaré domain Ω. Then for any p > n the following inequality holds   p(n−q) p p−n 1 p nq n |Ω  | n · ||Jϕ | L ∞ (Ω)||, ≤ inf B K (Ω)|Ω| p,q  μ p (Ω ) q∈(q ∗ ,n] where q ∗ = np/( p + n). Note that any convex domain Ω ⊂ Rn is the Sobolev–Poincaré domain and the constant Br ,q (Ω) can be estimated as [20] dn Br ,q (Ω) ≤ Ω n|Ω|

1− 1 n

−

1 q 1 q

+ +

1 r 1 r

!1− q1 + r1

1− n1

ωn

1

|Ω| n

− q1 + r1

,

n/2

where ωn = nΓ2π(n/2) is the volume of the unit ball in Rn and dΩ is the diameter of Ω. As examples, we consider non-convex star-shaped domains which are K quasiconformal ∞-regular domains. Example 1 The homeomorphism ϕ(x) = |x|a x, a > 0, is (a + 1)-quasiconformal and maps the n-dimensional cube Q := {xk ∈ Rn : |xk |
p2 > 1),

(12)

which can be found in [11] or [9, Ch.3]. Denoting, for 1 < q < ∞,

 2x  q1 1 q Aq (g; x) := |g(t)| dt , x x with natural A∞ (g; x) := ess sup |g(t)| for q = ∞, we can rewrite the norm in Oq , x≤t≤2x

1 < q ≤ ∞, as

 g Oq =

∞

Aq (g; x) d x.

0

In parallel to Oq we can introduce their subspaces E q as Eq =



f : f  ∈ Oq and



∞

0

|f(t)| dt < ∞ ; t

cf. [12, Ch. 2, Section 2.4]. Naturally, these spaces are endowed with the norm 



 f  E q =  f  Oq +

∞ 0

| f (t)| dt. t

The role of these spaces in the problems in questions is illustrated by the following result (see, e.g., [11]; there and in [12] as well as in some other sources the sine part is given in a more precise, asymptotic form). Theorem 4 Let f ∈ BV0 [0, ∞) ∩ L AC(0, ∞) and f  ∈ Oq (R+ ) for some 1 < q ≤ ∞. Then the cosine Fourier transform of f is integrable, with  f c  L 1 (R+ )   f   Oq (R+ ) ; Reprinted from the journal

243

E. Liflyand

if f ∈ E q (R+ ) for some 1 < q ≤ ∞, then the sine Fourier transform of f is integrable, with  f s  L 1 (R+ )   f  Eq (R+ ) . In [11] or [12], this theorem as well as its prototype for trigonometric series in [6] (see Theorem 6) were derived from more general results, mainly by means of the M. Riesz projection theorem. In [7], a direct proof of Theorem 4 can be found, based on the Hausdorff-Young inequality. Here and in what follows we use the notations “  ” and “  ” as abbreviations for “ ≤ C ” and “ ≥ C ”, with C being an absolute positive constant. 2.2 L1 convergence We are now in a position to formulate our main result. Since the approximated Fourier transform is supposed to be integrable, there is no need to approximate it near infinity, where it is already L 1 small. Therefore, a natural analog (and extension) of the periodic case reads as follows. Theorem 5 Let f ∈ BV0 [0, ∞)∩ L AC(0, ∞). If f  ∈ Oq (R+ ) for some 1 < q ≤ ∞, then, for any fixed M > 0, 

M

|S N ( f ; x)c −  f c (x)| d x = 0

lim

N →∞ 0

if and only if lim | f (t)| ln t = 0.

(13)

t→∞

If f ∈ E q (R+ ) for some 1 < q ≤ ∞, then, for any fixed M > 0, 

M

lim

N →∞ 0

|S N ( f ; x)s −  f s (x)| d x = 0

if and only if (13) holds true. Proof A typical way to establish the L 1 convergence for trigonometric series is to start with the Fejér means. We implement this expedient for the Fourier transforms as well. Here the Fejér means are  σ N (F; x)c =

N



N



0

t 1− N

 f (t) cos xt dt

and  σ N (F; x)s = 0

1−

t N

244

 f (t) sin xt dt.

Reprinted from the journal

L1 convergence of Fourier transforms

They are known to converge in L 1 norm. By this, we have to to search for conditions that ensure (14) lim S N ( f ; ·)c − σ N (F; ·)c  L 1 = 0, N →∞

and similarly for σ N (F; ·)s . Thus, (14) can be rewritten and reformulated as under what conditions    N   t  (15) f (t) cos t x dt  d x = 0, lim  N →∞ N R+

0

or analogously for the sine case, We present explicitly only the proof of the first part, since the difference in assumptions comes into play not in such a proof but in that of Theorem 4. More precisely, only  f   Oq will be involved in the following estimates. Taking into account the above arguments, we have to prove (15). First, we have 

1 N

0

   

N

f (t)

0

  1  N  N t cos t x dt  d x ≤ | f (t)| dt d x N 0 0  √N 1 | f (t)| dt + √ max | f (t)|, ≤ N 0 N ≤t≤N

which tends to zero as N → ∞. The estimate over [ N1 , M] is more delicate. Integrating by parts, we obtain  0

N

f (t)

 N  1 t cos t x − 1  t cos t x dt = f (t) sin t x +  N xN N x2 0  N  1 N  t cos t x − 1 − f (t) sin t x dt − f  (t) dt. x 0 N N x2 0

The integrated terms are  cos x N − 1 sin x := J1 + J2 , + f (N ) x N x2 

while the rest can be rewritten as 1 ix  −



N

−

f  (t) sin t x dt −



0 N 0

0

 

t sin t x dt f  (t) 1 − N

cos t x − 1 f  (t) dt := I1 + I2 . N x2

Estimating J1 leads to integrating over

1 N

≤ x ≤ M, which is equivalent to

 | f (N )|

Reprinted from the journal

N

M 1 N

245

dx , x

E. Liflyand

and letting this tend to zero is equivalent to (13). The same integration of J2 is equivalent to | f (N )|, which tends to zero by assumption. Let us proceed to estimating I1 and I2 . In I1 , there are two integrals. However, by the known properties of the Fourier transform of an integrable function (recall that  f  (x) f s uniformly. Since s x is integrable f  is integrable), both tend to the same limit  ( f  ∈ Q), we can change the passage to the limit as N → ∞ and integration, which makes I1 to tend to zero. The estimate of I2 obviously reduces to 1 N



M 1 N



− 32

x

√ | f  (t)| t dt d x.

N 0

Since for a nonnegative function φ, we have 

N

0

1 t





2t

φ(u) du dt = ln 2

t



N

2N

φ(u) du +

0

φ(u) ln

N

2N du, u

there holds 

N



√ | f  (t)| t dt 

0

N

1 √ t

0



2t

| f  (u)| du dt.

t

Applying Hölder’s inequality to the inner integral on the right, we obtain the bound, for q > 1, 

√

N

t Aq ( f  ; t) dt.

0

After integrating in x, we have to estimate 

1 √ N

N

√ t Aq ( f  ; t) dt.

0

Splitting the integral in two: over [0, is dominated by

√

√ N ] and over [ N , N ], we get that the first one 1

N − 4  f   Oq , which tends to zero as N → ∞. For the rest, we obtain 

N

√

Aq ( f  ; t) dt.

N

Since Aq ( f  ; t) is integrable, this integral tends to zero as N → ∞, which completes the proof.

246

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L1 convergence of Fourier transforms

3 Application to trigonometric series Starting from [11], most of the results on the integrability of trigonometric series were moved to proving the relevant estimates for the Fourier transforms, which can be further transferred to those for the series in the following way (more detailed description can be found in [12, Ch. 8]; more advanced way, for all the cases, not only those related to bounded variation, has recently been demonstrated in [13]). Given cosine and sine series with the null sequence of bv coefficients. Among numerous conditions for these trigonometric series to be a Fourier series, we concentrate on the sequence prototype of Theorem 4. For this, we need the prototypes of the Oq and E q spaces, written oq and eq , respectively. The first one will be defined, for 1 < q < ∞, by

doq =

∞ 

2

n q

n=0

⎧ ⎨2n+1 −1 ⎩

|dk |q

k=2n

and for q = ∞, by do∞ =

∞ 

⎫1 ⎬q ⎭

1 1 +  = 1, q q

< ∞,

sup |dn |.

(16)

(17)

k=0 n≤k 0 and Re (2μ + λ) > 0.) In relation to the system L D we will prove that the corresponding L p -Dirichlet problem in the upper half-space Rn+ with traces considered in the nontangential sense is not Fredholm solvable. This is made precise in Theorem 1.3. En route to it, we shall fully characterize the space of admissible boundary data for the L p -Dirichlet boundary problem for the system L D in the upper half-space in Theorem 1.1, and prove a structure theorem (cf. Theorem 1.2). This, among other things, shows that the L p -Dirichlet boundary problem for the system L D in the upper half-space has an infinite dimensional null-space (see (1.32), which should be compared with (1.4)). To state the aforementioned results, we need some notation. Given an aperture parameter κ > 0, the κ-nontangential cone with vertex at a point x = (x , 0) ∈ ∂Rn+ is

κ (x) := (y , t) ∈ Rn+ : |y − x | < κ t . Reprinted from the journal

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(1.11)

D. Mitrea et al.

Having fixed a function u : Rn+ → C and a point x ∈ Rn−1 ≡ ∂Rn+ , the κnontangential boundary limit of u at x is defined as  κ−n.t.  u  n (x ) := ∂R+

lim

(y ,t)∈κ (x ,0) (y ,t)−→(x ,0)

u(y , t),

(1.12)

whenever the limit exists. Also, the κ-nontangential maximal function of u at a point x ∈ Rn−1 is   Nκ u (x ) := esssup(y ,t)∈κ (x ,0) |u(y , t)|,

(1.13)

where the essential supremum is taken with respect to Ln , the n-dimensional Lebesgue measure. Consequently, whenever the κ-nontangential limit of u at x exists we have  κ−n.t.         u ∂ (x ) ≤ Nκ u (x ).

(1.14)

It is also useful to note that if κ > 0 is another aperture parameter and p ∈ (1, ∞), then there exist two finite constants C0 , C1 > 0, which depend only on n, κ, κ , and p, such that C0 Nκ u L p (Rn−1 ) ≤ Nκ u L p (Rn−1 ) ≤ C1 Nκ u L p (Rn−1 ) ,

(1.15)

for each Lebesgue measurable function u : Rn+ → C. See [9, § 2.5.1, p. 62] For each p ∈ (1, ∞) we then consider the L p -Dirichlet boundary problem for the system L D in the upper half-space: ⎧  n u ∈ C ∞ (Rn+ ) , ⎪ ⎪ ⎪ ⎨  u − 2∇div u = 0 in Rn+ , Nκ u ∈ L p (Rn−1 ), ⎪ ⎪ κ−n.t. ⎪ ⎩ u n = f ∈  L p (Rn−1 )n .

(1.16)

∂R+

Our first result provides a complete description for the space of admissible boundary data for the Dirichlet Problem (1.16). In order to state it, we recall the definition of the Riesz transforms. Specifically, for each index j ∈ {1, . . . , n − 1} define the (ordinary) j-th Riesz transform in Rn−1 as the singular integral operator R j acting on any given   function f belonging to the weighted space L 1 Rn−1 , 1+|xd x |n−1 according to

R j f (x ) := lim

ε→0+

2 ωn−1

 y ∈Rn−1 |x −y |>ε

x j − yj f (y ) dy for Ln−1 − a.e. x ∈ Rn−1 . |x − y |n (1.17) 254

Reprinted from the journal

Failure of Fredholm solvability...

We recall that for each index j ∈ {1, . . . , n − 1}, and each integrability exponent p ∈ (1, ∞), the operator R j : L p (Rn−1 ) → L p (Rn−1 ) is well defined, linear, and bounded.

(1.18)

Here is the characterization of the space of admissible boundary data for the L p Dirichlet boundary problem for the system L D in the upper half-space, alluded to earlier. Theorem 1.1 Fix an integer n ∈ N with n ≥ 2, an integrability exponent p ∈ (1, ∞), and an aperture parameter κ ∈ (0, ∞). Then the space of admissible boundary data for the L p -Dirichlet Problem for the system L D in the upper half-space  κ−n.t.  n u∂Rn : u ∈ C ∞ (Rn+ ) , L D u = 0 in Rn+ , Nκ u ∈ L p (Rn−1 ), and +  κ−n.t. u∂Rn exists Ln−1 − a.e. in Rn−1 +

(1.19) may be described as 

 n−1   n Rj fj . f = ( f 1 , . . . , f n ) ∈ L p (Rn−1 ) : f n = −

(1.20)

j=1

The following structure theorem is a key ingredient in the proof of our main result, stated in Theorem 1.3. Theorem 1.2 Fix an integer n ∈ N with n ≥ 2, an integrability exponent p ∈ (1, ∞), and an aperture parameter κ ∈ (0, ∞). Then any vector-valued function u satisfying ⎧  n ⎨ u ∈ C ∞ (Rn+ ) ,  u − 2∇div u = 0 in Rn+ , ⎩ Nκ (∇ u) ∈ L p (Rn−1 ),

(1.21)

is of the form u(x) = v(x) + xn (∇w)(x), ∀ x = (x , xn ) ∈ Rn−1 × (0, ∞),

(1.22)

for some scalar function w satisfying ⎧ ⎨ w ∈ C ∞ (Rn+ ), w = 0 in Rn+ , ⎩ Nκ (∇w) ∈ L p (Rn−1 ), Reprinted from the journal

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(1.23)

D. Mitrea et al.

and some vector-valued function v satisfying ⎧  n ⎨ v ∈ C ∞ (Rn+ ) ,  v = 0, div v = 0, in Rn+ , ⎩ Nκ (∇ v) ∈ L p (Rn−1 ).

(1.24)

κ−n.t. Moreover, the nontangential boundary trace u∂Rn exists Ln−1 -a.e. on Rn−1 , and + whenever the functions w, v are as in (1.22–1.24), the nontangential boundary trace κ−n.t. v∂Rn exists Ln−1 -a.e. on Rn−1 and one has +

κ−n.t. κ−n.t. v∂Rn = u∂Rn . +

(1.25)

+

Conversely, for each w as in (1.23) and each v as in (1.24), the vector-valued function u associated with w and v as in (1.22) has the properties listed in (1.21) and satisfies (1.25). Furthermore, if Nκ w ∈ L p (Rn−1 ) and Nκ v ∈ L p (Rn−1 ) then Nκ u ∈ L p (Rn−1 ).

(1.26)

A few important consequences of this structure theorem are as follows. First, as a result of Theorem 1.2 we have that for each scalar function w ∈ C ∞ (Rn+ ) with w = 0 in Rn+ and Nκ (∇w) ∈ L p (Rn−1 ),

(1.27)

the vector-valued function u : Rn+ −→ Cn , u(x) := xn (∇w)(x) for each x = (x1 , . . . , xn ) ∈ Rn+ , (1.28) satisfies  n u ∈ C ∞ (Rn+ ) , L D u = 0 in Rn+ , Nκ (∇ u) ∈ L p (Rn−1 ), κ−n.t. and u∂Rn = 0 at Ln−1 − a.e. point on Rn−1 .

(1.29)

+

In the converse direction, each vector-valued function u as in (1.29) has the format described in (1.28) for some scalar function w as in (1.27) (this is seen from (1.25) and the uniqueness in the Homogeneous Regularity Problem for the Laplacian in the upper-half space; cf. [4]). In addition, from the last part in Theorem 1.2 we see that if in place of (1.27) one now assumes w ∈ C ∞ (Rn+ ) with w = 0 in Rn+ and Nκ w ∈ L p (Rn−1 ), Nκ (∇w) ∈ L p (Rn−1 ), 256

(1.30)

Reprinted from the journal

Failure of Fredholm solvability...

then the vector-valued function u defined as in (1.28) for this choice of w has the additional property that Nκ u ∈ L p (Rn−1 ), i.e., satisfies  n u ∈ C ∞ (Rn+ ) , L D u = 0 in Rn+ , Nκ u, Nκ (∇ u) ∈ L p (Rn−1 ), κ−n.t. (1.31) and u∂Rn = 0 at Ln−1 − a.e. point on Rn−1 . +

In particular, the space of null-solutions for the Dirichlet Problem(1.16) for the system L D in the upper half-space is infinite dimensional. (1.32) Since the intersection of the space described in (1.20) with the infinite dimensional vector space (0, . . . , 0, f ) : f ∈ L p (Rn−1 ) is the singleton {(0, . . . , 0)}, from Theorem 1.1 and (1.32) we arrive at the following result. Theorem 1.3 Fix n ∈ N with n ≥ 2 and let p ∈ (1, ∞) be an arbitrary integrability exponent. Then the Dirichlet Problem (1.16) is not Fredholm solvable. In fact, problem (1.16) has an infinite dimensional space of null-solutions n and its space of admissible boundary data has infinite codimension in L p (Rn−1 ) .

2 Preliminaries As a preamble to the proofs of the results stated in §1, in this section we take care of a number of preliminary results. We begin by recalling a suitable version of the classical Divergence Theorem, proved in [8] (see also [7]). Theorem 2.1 Fix n ∈ N with n ≥ 2 and pick some arbitrary aperture parameter κ ∈ (0, ∞). Assume the vector field F = (F1 , . . . , Fn ) : Rn+ → Cn , with Lebesgue measurable components, satisfies the following properties: κ−n.t. the nontangential trace Fn ∂Rn exists Ln−1 − a.e. in ∂Rn+ ≡ Rn−1 , + Nκ F ∈ L 1 (Rn−1 ), and div F := ∂1 F1 + · · · + ∂n Fn ∈ L 1 (Rn+ ),

(2.1)

where all partial derivatives are considered in the sense of distributions in Rn+ . κ −n.t. Then for any other aperture parameter κ > 0 the nontangential trace Fn ∂Rn +

exists at Ln−1 -a.e. point on Rn−1 and is actually independent of κ . When regarding it as a function defined Ln−1 -a.e. in Rn−1 , this belongs to L 1 (Rn−1 ) and, with the dependence on the parameter κ dropped,    n.t.  Fn ∂Rn dLn−1 . div F dLn = − (2.2) Rn+

Rn−1

+

1,1 In what follows, Wloc (Rn+ ) denotes the L 1 -based local Sobolev space of order one p in Rn+ . Also, for each p ∈ [1, ∞], we denote by L 1 (Rn−1 ) the L p -based Sobolev

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space of order one in Rn−1 , while L 1,loc (Rn−1 ) denotes the space of locally p-th power integrable functions in Rn−1 with first order derivatives locally p-th power integrable in Rn−1 . Lemma 2.2 Fix an aperture parameter κ > 0 along with a truncation parameter ρ 1,1 (Rn+ ) is a function with the property that Nκ ω and ρ ∈ (0, ∞). Suppose ω ∈ Wloc  κ−n.t. ρ Nκ (∇ω) belong to L 1loc (Rn−1 ) and the nontangential boundary trace ω∂Rn exists at + κ−n.t. Ln−1 -a.e. point in Rn−1 ≡ ∂Rn+ . If also (∂ j ω)∂Rn exists at Ln−1 -a.e. point in Rn−1 + for some j ∈ {1, . . . , n − 1}, then κ−n.t.  κ−n.t.  ∂ j ω∂Rn = (∂ j ω)∂Rn in D (Rn−1 ). +

(2.3)

+

κ−n.t. κ−n.t. Consequently, if (∇ω)∂Rn exists at Ln−1 -a.e. point in Rn−1 , then ω∂Rn ∈ +

+

L 11,loc (Rn−1 ). ρ ρ Moreover, if in addition Nκ ω, Nκ (∇ω) ∈ L p (Rn−1 ) for some p ∈ (1, ∞), then κ−n.t. p ω∂Rn ∈ L 1 (Rn−1 ). +

κ−n.t. κ−n.t. Proof From [8] we know that ω∂Rn and (∂ j ω)∂Rn are Ln−1 -measurable functions. + + Bearing this in mind, we conclude from (1.14) and assumptions that, in fact, κ−n.t. κ−n.t. ω∂Rn and (∂ j ω)∂Rn belong to L 1loc (Rn−1 ). +

+

(2.4)

Fix an arbitrary test function ϕ ∈ Cc∞ (Rn−1 ). Using the definition of distributional derivatives and (2.4), write D (Rn−1 )

 κ−n.t. 

κ−n.t.

∂ j ω∂Rn , ϕ D(Rn−1 ) = −D (Rn−1 ) ω∂Rn , ∂ j ϕ D(Rn−1 ) + +   κ−n.t.  ω n (∂ j ϕ) dLn−1 . =− Rn−1

∂R+

(2.5)

Now pick a function  satisfying   ∈ Cc∞ (Rn ), Rn−1 ×{0} = ϕ,

supp  ⊆ (x , xn ) ∈ Rn−1 × R : |xn | < ρ ,

(2.6)

and note that, since j < n, we have (∂ j )(x , 0) = (∂ j ϕ)(x ) for each x ∈ Rn−1 .

(2.7)

Let e : 1 ≤  ≤ n denote the standard orthonormal basis in Rn . Consider the vector field defined at Ln -a.e. point in Rn+ as     F := ∂ j (ω )en − ∂n (ω )e j = ω ∂ j  + (∂ j ω) en − ω ∂n  + (∂n ω) e j . (2.8) 258

Reprinted from the journal

Failure of Fredholm solvability...

κ−n.t.  n  n exists at Ln−1 -a.e. Then F has Lebesgue measurable components, the trace ( F) ∂R +

point in Rn−1 and, in fact, κ−n.t. κ−n.t.  κ−n.t.   n  n = ω n ∂ j ϕ + (∂ j ω) n ϕ, ( F) ∂R ∂R ∂R +

+

(2.9)

+

thanks to (2.6), (2.7), (2.8), and assumptions. Moreover, from (2.8) and the second line in (2.6) it follows that     Nκ F ≤ Nκρ ω · Nκ (∇) + Nκρ (∇ω) · Nκ  . ρ

(2.10)

ρ

Since Nκ ω, Nκ (∇ω) ∈ L 1loc (Rn−1 ) while Nκ , Nκ (∇) belong to L ∞ (Rn−1 ) and have compact support, from (2.10) we conclude that Nκ F ∈ L 1 (Rn−1 ). In addition, the first equality in (2.8) implies that div F = 0 in D (Rn+ ). We may therefore apply Theorem 2.1 to compute  Rn−1

  κ−n.t.  κ−n.t.  ω∂Rn ∂ j ϕ + (∂ j ω)∂Rn ϕ dLn−1 = +

+

Rn−1

=−



κ−n.t.  n  n dLn−1 ( F) ∂R

Rn+

+

div F dLn = 0.

(2.11)

Together, (2.5) and (2.11) imply  κ−n.t. 

 D (Rn−1 ) ∂ j ω ∂Rn+ , ϕ D(Rn−1 ) =

 Rn−1

κ−n.t. (∂ j ω)∂Rn ϕ dLn−1 +

κ−n.t.

= D (Rn−1 ) (∂ j ω)∂Rn , ϕ D(Rn−1 ) . +

(2.12)

The desired conclusion follows from (2.12) and the arbitrariness of ϕ in Cc∞ (Rn−1 ).   Moving on, we introduce the Dirichlet-to-Neumann operator. To define it, we first recall the Poisson kernel for the Laplacian P  (x ) :=

2 ωn−1

·

1 for all x ∈ Rn−1 , (1 + |x |2 )n/2

(2.13)

where ωn−1 is the surface area of the unit sphere in Rn , and Pt := t 1−n P  (·/t) for each t > 0. The action of the Dirichlet-to-Neumann operator from a Sobolev space of order one into the corresponding Lebesgue space is discussed in the next lemma. Lemma 2.3 Having fixed an integrability exponent p ∈ (1, ∞), consider the Dirichletto-Neumann operator p

DN : L 1 (Rn−1 ) −→ L p (Rn−1 ) Reprinted from the journal

259

(2.14)

D. Mitrea et al. p

acting on each f ∈ L 1 (Rn−1 ) according to       DN f (x ) := lim ∂t Pt ∗ f (x ) for Ln−1 − a.e. x ∈ Rn−1 . (2.15) t→0+

Then this is a well-defined, linear, bounded, and injective operator. In addition, p given any aperture parameter κ > 0, for each f ∈ L 1 (Rn−1 ) one has κ−n.t. DN f = (∂n u)∂Rn

(2.16)

+

where u is the unique solution of the Regularity Problem for the Laplacian in Rn+ with boundary datum f , formulated as ⎧ u ∈ C ∞ (Rn+ ), ⎪ ⎪ ⎪ ⎨ u = 0 in Rn+ , Nκ u, Nκ (∇u) ∈ L p (Rn−1 ), ⎪ ⎪ κ−n.t. ⎪ ⎩ u  n = f at Ln−1 − a.e. point on Rn−1 . ∂R

(2.17)

+

The proof of this lemma uses certain properties of layer potential operators associated with the upper half-space which are discussed next. Let E  denote the standard fundamental solution for the Laplacian in Rn , i.e., ⎧ ⎪ ⎨

1 1 if n ≥ 3, n−2 ω (2 − n) |x| n−1 E  (x) := (2.18) ⎪ ⎩ 1 ln |x| if n = 2, 2π   for each x ∈ Rn \ {0}. Fix an arbitrary function f ∈ L 1 Rn−1 , 1+|xd x |n−1 . Then the harmonic double layer D for the upper half-space acting on f is defined by   D f (x , t) := 2−1 Pt ∗ f (x ) for all (x , t) ∈ Rn−1 × (0, ∞),

(2.19)

while the modified harmonic single layer Smod for the upper half-space acting on f is defined at each x = (x , xn ) ∈ Rn−1 × (0, ∞) by



Smod f (x , xn ) :=

Rn−1



E  (x − y , xn ) − E  (−y , 0)1Rn \B(0,1) (−y , 0) f (y ) dy . (2.20)

The main properties of the modified harmonic single layer Smod which are relevant for this work are as follows. If p ∈ (1, ∞), κ ∈ (0, ∞), and f ∈ L p (Rn−1 ) then   Smod f ∈ C ∞ (Rn+ ),  Smod f = 0 in Rn+ ,   Nκ (∇(S f )) p n−1 ≤ C f L p (Rn−1 ) , mod

L (R

)

260

(2.21) (2.22) Reprinted from the journal

Failure of Fredholm solvability...

κ−n.t. (∂n Smod f )∂Rn = +

1 2

f at Ln−1 − a.e. point in Rn−1 ,

(2.23)

where the constant C = C(n, p, κ) ∈ (0, ∞) is independent of f . p

Proof of Lemma 2.3 Fix an arbitrary function f ∈ L 1 (Rn−1 ) and pick some aperture parameter κ > 0. Recall from [5] that the function   u(x , t) := Pt  ∗ f (x ) for all x ∈ Rn−1 , t > 0,

(2.24)

is the unique solution of (2.17). Consequently, the expression in (2.15) may be written in terms of u as κ−n.t. DN f = (∂n u)∂Rn ∈ L p (Rn−1 ). +

(2.25)

This proves that the Dirichlet-to-Neumann operator defined as in (2.14–2.15) is well defined, linear, bounded, and that (2.16) holds. To show that the Dirichlet-to-Neumann p operator is also injective, pick f ∈ L 1 (Rn−1 ) such that DN f = 0. Then u as in (2.24) κ−n.t. satisfies (∂n u) n = 0. Based on the integral representation formula for the Neumann ∂R+

Problem from [8] we obtain that u is of the form u = D f + c in Rn+ ,

(2.26)

for some c ∈ C. Taking the nontangential trace in (2.26), and observing that  κ−n.t. D f ∂Rn = 21 f , we arrive at f = 21 f + c, thus f is constant in Rn−1 . Upon +

recalling that f ∈ L p (Rn−1 ), we conclude that f = 0. Hence DN is indeed injective in the context of (2.14). Since the Riesz transforms are convolution-type singular integral operators, in addition to (1.18) we have p

p

R j : L 1 (Rn−1 ) → L 1 (Rn−1 ) is well defined, linear, and bounded

(2.27)

for each j ∈ {1, . . . , n − 1} and each p ∈ (1, ∞). In relation to this, it is useful to know that each partial derivative factors as the composition between the Dirichlet-toNeumann map and the corresponding Riesz transform. p

Lemma 2.4 For each f ∈ L 1 (Rn−1 ) with p ∈ (1, ∞) and each j ∈ {1, . . . , n − 1} one has DN (R j f ) = ∂ j f at Ln−1 − a.e. point in Rn−1 , where R j is the j-th Riesz transform in Rn−1 (cf. (1.17) and (2.27)). Proof Fix j ∈ {1, . . . , n − 1} and define Reprinted from the journal

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(2.28)

D. Mitrea et al.

u(x) :=

2 ωn−1

 Rn−1

x j − yj f (y ) dy , ∀ x = (x1 , . . . , xn ) ∈ Rn+ . |x − (y , 0)|n (2.29)

Then for each k ∈ {1, . . . , n} we may integrate by parts and obtain  (∂k u)(x) = −2  =2

Rn−1

Rn−1

  ∂ y j (∂k E  )(x − (y , 0)) f (y ) dy

(∂k E  )(x − (y , 0)) (∂ j f )(y ) dy

= 2∂k Smod (∂ j f )(x), ∀ x ∈ Rn+ .

(2.30)

Then, as seen from (2.29), (2.30), (2.22), and standard Calderón-Zygmund theory, u ∈ C ∞ (Rn+ ), u = 0 in Rn+ , Nκ u, Nκ (∇u) ∈ L p (Rn−1 ), and κ−n.t. u  n = R j f at Ln−1 − a.e. point in Rn−1 . ∂R+

(2.31)

Hence, we may invoke (2.16), (2.30), and (2.23) to write κ−n.t. κ−n.t. DN (R j f ) = (∂n u)∂Rn = 2(∂n Smod )(∂ j f )∂Rn = ∂ j f +

+

at Ln−1 -a.e. point in Rn−1 .

(2.32)  

The next step in our program is to describe the space of admissible boundary data for the Dirichlet Problem for vector-valued diverge-free harmonic functions in the upper half-space. Proposition 2.5 Fix an aperture parameter κ > 0 and an integrability exponent p ∈ (1, ∞). Then  κ−n.t.   n v∂Rn : v ∈ C ∞ (Rn+ ) ,  v = 0 and div v = 0 in Rn+ , Nκ v ∈ L p (Rn−1 ) +



=

 n−1   n Rj fj , f = ( f 1 , . . . , f n ) ∈ L p (Rn−1 ) : f n = −

(2.33)

j=1

where the R j ’s with j ∈ {1, . . . , n − 1} are the Riesz transforms in Rn−1 (cf. (1.17)).  n v = 0 and div v = 0 in Proof Let v = (v j )1≤ j≤n ∈ C ∞ (Rn+ ) be such that  Rn+ , and Nκ v belongs to L p (Rn−1 ). Then Calderón’s theorem (see [3]) ensures the κ−n.t. existence of the nontangential trace v n at Ln−1 -a.e. point in Rn−1 . Denote ∂R+

κ−n.t. f := ( f j )1≤ j≤n := v∂Rn . +

262

(2.34) Reprinted from the journal

Failure of Fredholm solvability...

Fix ε > 0 and define    n  as well as vε := v · +εen ∈ C ∞ Rn+  (ε) f j := v j (· + εen )∂Rn for each j ∈ {1, . . . , n}.

(2.35)

 vε = 0 and div vε = 0 in Rn+ .

(2.36)

+

Then

Since κ (x) + εen ⊆ κ (x) for each x ∈ ∂Rn+ , we also have Nκ vε ≤ Nκ v, which implies Nκ vε ∈ L p (Rn−1 ).

(2.37)

Moreover, by Lebesgue’s Dominated Convergence Theorem (with uniform domination provided by Nκ v ∈ L p (Rn−1 )), we obtain (ε)

fj

→ f j in L p (Rn−1 ) as ε → 0+ , for each j ∈ {1, . . . , n}.

(2.38)

Simple geometric considerations imply that there exists  κ > κ with the property that for each z ∈ ∂Rn+ and each x = (x , xn ) ∈ κ (z) there holds B(x, xn /2) ⊆ κ (z).

(2.39)

In particular, if z ∈ ∂Rn+ and

x = (x , xn ) ∈ κ (z) has xn > ε then B(x, ε/2) ⊆ κ (z).

(2.40)

Hence, if z ∈ ∂Rn+ and x ∈ κ (z) we may use interior estimates for the harmonic function v in Rn+ to write         ∇ vε (x) =  ∇ v (x + εen ) ≤ C − | v | dLn ε B(x+εen , ε/2)   ≤ Cε Nκ v (z).

(2.41)

Now taking the supremum over all x ∈ κ (z) in the resulting estimate in (2.41) we obtain     Nκ ∇ vε (z) ≤ Cε Nκ v (z) for all z ∈ ∂Rn+ ,

(2.42)

  Nκ ∇ vε ∈ L p (Rn−1 ).

(2.43)

thus

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D. Mitrea et al.

  This, (2.37), the fact that the pointwise trace ∇v j (· + εen ) ∂Rn exists for each + j ∈ {1, . . . , n}, (2.35), and Lemma 2.2 then yield 

(ε) 

fj

1≤ j≤n

  p n = vε ∂Rn ∈ L 1 (Rn−1 ) .

(2.44)

+

Furthermore, at Ln−1 -a.e. point in Rn−1 we have (ε)

(ε)

(ε)

∂1 f 1 + ∂2 f 2 + · · · + ∂n−1 f n−1 = (∂1 v1 )(· + εen ) + · · · + (∂n−1 vn−1 )(· + εen )     = (div v )(· + εen ) − (∂n vn )(· + εen ) = −∂n vn (· + εen ) = −DN f n(ε) , (2.45) where the last equality uses (2.16). In addition, from Lemma 2.4 we know that ∂ j f j(ε) = DN (R j f j(ε) ) at Ln−1 − a.e. point in Rn−1 , ∀ j ∈ {1, . . . , n − 1}. (2.46) Together, (2.46) and (2.45) imply DN

 n−1

R j f j(ε)



  = −DN f n(ε) at Ln−1 − a.e. point in Rn−1 .

(2.47)

j=1

Since DN is injective (cf. Lemma 2.3), it follows that n−1 

R j f j(ε) = − f n(ε) at Ln−1 − a.e. point in Rn−1 .

(2.48)

j=1

By combining (2.48), (2.38), and the continuity of the Riesz transforms on L p (Rn−1 ), n−1 -a.e. point in Rn−1 . This we arrive at the conclusion that n−1 j=1 R j f j = − f n at L finishes the proof of the left-to-right inclusion in (2.33). In the converse direction, suppose n−1   n ( f 1 , . . . , f n ) ∈ L p (Rn−1 ) are such that f n = − Rj f j.

(2.49)

j=1

Pick an arbitrary φ ∈ C ∞ (Rn−1 ) with

 Rn−1

φ dLn−1 = 1 and for each ε > 0 set

φε (x ) := ε1−n φ(x /ε), ∀ x ∈ Rn−1 .

(2.50)

Also, for ε > 0, define (ε)

fj

:= φε ∗ f j in Rn−1 , for each j ∈ {1, . . . , n}. 264

(2.51)

Reprinted from the journal

Failure of Fredholm solvability...

Given the current assumptions, for each j ∈ {1, . . . , n} we then have (ε)

fj

(ε)

p

∈ L 1 (Rn−1 ) and f j

→ f j in L p (Rn−1 ) as ε → 0+ .

(2.52)

Moreover, f n(ε) = φε ∗ f n = −

n−1 

φε ∗ R j f j .

(2.53)

j=1

We make the claim that   φε ∗ (R j f j ) = R j f j(ε) for each j ∈ {1, . . . , n}.

(2.54)

To see why this is true, denote by F the Fourier transform in Rn−1 , and for each ξ ∈ Rn−1 \ {0}, using the properties of the Fourier transform, compute       F φε ∗ (R j f j ) (ξ ) = F φε (ξ )F R j f j (ξ )     ξj = F φε (ξ )(−i) F f j (ξ ) |ξ | ξ j     = −i F φε (ξ )F f j (ξ ) |ξ |    ξj (ε)  = −i F φε ∗ f j (ξ ) = F R j ( f j ) (ξ ). |ξ |

(2.55)

Now (2.54) follows by applying the inverse Fourier transform to the resulting identity in (2.55). Combining (2.53) and (2.54) we arrive at f n(ε) = −

n−1 

 (ε)  R j f j , ∀ ε > 0.

(2.56)

j=1

Next, for each ε > 0 define vε (x , t) :=



(ε) 

Pt ∗ f j

(x )

 1≤ j≤n

, ∀ x ∈ Rn−1 , ∀ t > 0,

(2.57)

and v(x , t) :=



  Pt ∗ f j (x )

1≤ j≤n

, ∀ x ∈ Rn−1 , ∀ t > 0.

(2.58)

From the properties of the Poisson kernel for the Laplacian, these definitions, and (2.49), it follows that  n v, vε ∈ C ∞ (Rn+ ) , Reprinted from the journal

(2.59) 265

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 v = 0 and  vε = 0 in Rn+ ,   Nκ v, Nκ vε , Nκ ∇ vε ∈ L p (Rn−1 ), κ−n.t.   v∂Rn = f j 1≤ j≤n at Ln−1 − a.e. point in Rn−1 . +

(2.60) (2.61) (2.62)

In order to conclude that v is contained in the space appearing in the left-hand side of (2.33), there remains to show that div v = 0 in Rn+ . To this end, we make two claims: vε → v uniformly on compact sets in Rn+ as ε → 0+ ,

(2.63)

and div vε = 0 in

(2.64)

Rn+ ,

for each ε > 0.

Assume for now that (2.63–2.64) are true. Then for each ϕ ∈ Cc∞ (Rn+ ) we may write  D (Rn+ )

div v , ϕ

D(Rn+ )

=−

[D (Rn+ )]n

 v , ∇ϕ

[D (Rn+ )]n



= − lim

ε→0+ Rn+

=−

Rn+

v · ∇ϕ dLn



vε · ∇ϕ dLn = lim

ε→0+ Rn+

(div vε )ϕ dLn = 0. (2.65)

In turn, (2.65) implies div v = 0 in D (Rn+ ), which further yields div v = 0 pointwise in Rn+ , as wanted. To complete the proof of the proposition we are left with justifying (2.63) and (2.64). We will first prove (2.63). Let K be a compact set in Rn+ . Then there exists ε0 > 0 with the property that

K ⊆ (x , t) ∈ Rn+ : x ∈ Rn−1 , t > ε0 .

(2.66)

If g ∈ L p (Rn−1 ) is arbitrary, we may use Hölder’s inequality to estimate      sup  Pt ∗ g (x ) ≤

(x ,t)∈K

sup

(x ,t)∈K

    g L p (Rn−1 )  Pt  L p (Rn−1 )

= g L p (Rn−1 )

  sup t −(n−1)/ p  P   L p (Rn−1 )

(x ,t)∈K

≤ C(n, p, ε0 ) g L p (Rn−1 ) ,

(2.67)

where p := (1 − 1/ p)−1 . The estimate in (2.67) may then be used to write n    (ε) f j − f j L p (Rn−1 ) , ∀ ε > ε0 . sup vε − v ≤ C(n, p, ε0 ) K

(2.68)

j=1

Together, (2.68) and (2.52) yield (2.63). 266

Reprinted from the journal

Failure of Fredholm solvability...

As for (2.64), start with (2.57), then invoke (2.15), (2.16), and Lemma 2.4 to write n−1        κ−n.t.  (ε)  div vε ∂Rn (x , 0) = lim Pt ∗ (∂ j f j ) (x ) + lim ∂t Pt ∗ f n(ε) (x ) +

j=1

t→0+

t→0+

n−1    = (∂ j f j(ε) )(x ) + DN f n(ε) (x ) j=1

=

n−1 

    DN R j f j(ε) (x ) + DN f n(ε) (x ) = 0,

(2.69)

j=1

at Ln−1 -a.e. point x ∈ Rn−1 , where the last equality employs (2.56). Since we also know that div vε ∈ C ∞ (Rn+ ) is harmonic in Rn+ and     Nκ div vε ≤ CNκ ∇ vε ∈ L p (Rn−1 ),

(2.70)

now (2.64) follows by invoking the uniqueness of solution for the L p -Dirichlet problem   for the Laplacian in Rn+ (see, e.g., [5]).

3 Proofs of Theorem 1.2 and Theorem 1.1 First we prove the structure theorem stated in §1. Proof of Theorem 1.2 For starters, observe that thanks to Calderón’s theorem [3], κ−n.t. if w satisfies (1.23) then (∇w)∂Rn exists a.e. +  n and belongs to L p (Rn−1 ) .

(3.1)

Also, since the Mean Value Formula gives that v is Lipschitz in each κ (x) with x ∈ ∂Rn+ such that Nκ (∇ v)(x) < +∞, we conclude that κ−n.t. if v satisfies (1.24) then v∂Rn exists a.e. in Rn−1 . +

(3.2)

Now suppose u is as in (1.21). Then κ−n.t. κ−n.t. u ∂Rn and (∇u)∂Rn exist Ln−1 − a.e. in Rn−1 , +

+

(3.3)

where the existence of the first nontangential trace is a consequence of the Mean Value Formula (reasoning as above), and the existence of the second nontangential trace is implied by the Fatou-type result established in [8]. Furthermore, ⎧ ⎨ div u ∈ C ∞ (Rn+ ), (div u) = 0 in Rn+ , (3.4) ⎩ Nκ (div u) ∈ L p (Rn−1 ). Reprinted from the journal

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In addition, Calderón’s theorem [3] implies that κ−n.t. (div u)∂Rn exists at Ln−1 − a.e. point in Rn−1 and belongs to L p (Rn−1 ). +

(3.5) Next, introduce  κ−n.t.  in Rn+ . w := 2Smod (div u)∂Rn

(3.6)

+

The properties of the operator Smod recorded in (2.21–2.23) imply ⎧ n ∞ = 0 in Rn+ , ⎪ ⎨ w ∈ C (R+ ),p w n−1 Nκ ∇w ∈ L (R ),  κ−n.t.  κ−n.t. ⎪ ⎩ ∂n w  n = div u  n . ∂R+

(3.7)

∂R+

Hence, w satisfies all properties listed in (1.23), and ∂n w is a solution of the L p Dirichlet boundary value problem for the Laplacian in Rn+ with boundary datum κ−n.t. (div u) n . In view of (3.4), uniqueness for this problem then implies ∂R+

∂n w = div u in Rn+ ,

(3.8)

while (3.6) and (2.22) give κ−n.t.   Nκ (∇w) L p (Rn−1 ) ≤ C (div u)∂Rn  L p (Rn−1 ) ≤ C Nκ (∇ u) L p (Rn−1 ) . +

(3.9)

To proceed, define v(x) := u(x) − xn (∇w)(x), ∀ x ∈ Rn+ .

(3.10)

Then (1.21), (3.7), (3.1), (3.3), and (3.8) guarantee that κ−n.t. κ−n.t. n  v ∈ C ∞ (Rn+ ) , v∂Rn = u∂Rn , and + + div v = div u − ∂n w = 0 in Rn+ .

(3.11)

Also, using (3.10) and (3.8) we compute ( v )(x) = ( u )(x) − 2∇(∂n w)(x) − xn ∇(w)(x) = ( u )(x) − 2∇(div u)(x) = 0 for each x ∈ Rn+ .

(3.12)

Let  κ be as in (2.39). Then we may use interior estimates for the harmonic function ∇w in Rn+ to estimate    xn (∇ 2 w)(x) ≤ cn −

B(x,xn /2)

|∇w| dLn

268

Reprinted from the journal

Failure of Fredholm solvability...

  ≤ cn Nκ ∇w (z), ∀ x = (x , xn ) ∈ κ (z),

(3.13)

where cn ∈ (0, ∞) depends only on n. Together, (3.10) and (3.13) yield       Nκ ∂ j v (z) ≤ cn Nκ ∇ u (z) + cn Nκ ∇w (z), for all j ∈ {1, . . . , n} and all z ∈ ∂Rn+ .

(3.14)

Collectively, (3.14), (3.9), and (1.15) then imply     Nκ ∇ v L p (Rn−1 ) ≤ C Nκ ∇ v L p (Rn−1 )     ≤ C Nκ ∇ u L p (Rn−1 ) + C Nκ ∇w L p (Rn−1 )   ≤ C Nκ ∇ u L p (Rn−1 ) , (3.15) for some C = C(n, κ, κ , p) ∈ (0, by assumption Nκ (∇ u) ∈ L p (Rn−1 ),  ∞).  Since p n−1 we ultimately conclude that Nκ ∇ v ∈ L (R ). Together with (3.11) and (3.12), this shows that v satisfies (1.24). Recalling (3.10), it follows that u is of the form (1.22) and that (1.25) holds. To prove the converse statement, assume w is as in (1.23), v is as in (1.24), anddefine n u associated with w and v as in (1.22). Then it is immediate that u ∈ C ∞ (Rn+ ) and   div xn (∇w)(x) = (∂n w)(x) + xn (w)(x) = (∂n w)(x) for all x ∈ Rn+ . (3.16) In view of (3.16) and the properties of v and w, we obtain  u − 2∇div u

  =  v + (xn )∇w + 2∇(∂n w) + xn ∇(w) − 2∇div v − 2∇ div(xn ∇w) = 2∇(∂n w) − 2∇[∂n w] = 0 in Rn+ . (3.17)

In addition, for each j ∈ {1, . . . , n}, after applying ∂ j in (1.22) and using (2.39), (3.13), we may estimate       Nκ ∂ j u (z) ≤ cn Nκ ∇ v (z) + cn Nκ ∇w (z), ∀ z ∈ ∂Rn+ .

(3.18)

Together with  the  memberships in the last lines in (1.23) and (1.24), the above estimate   implies Nκ ∇ u ∈ L p (Rn−1 ), which when combined with (1.15) gives Nκ ∇ u ∈ L p (Rn−1 ). We have thus shown that u satisfies (1.21). That (1.25) also holds, follows from (1.22) and (3.1–3.2). Finally, make the additional assumptions that Nκ w and Nκ v belong to L p (Rn−1 ). Interior estimates for the harmonic function w in Rn+ together with (2.39) imply that for each z ∈ ∂Rn+ we have    xn (∇w)(x) ≤ cn −

Reprinted from the journal

B(x,xn /2)

269

|w| dLn ≤ cn Nκ w(z)

(3.19)

D. Mitrea et al.

at each point x = (x , xn ) ∈ κ (z). In concert with (1.22) this shows that the estimate Nκ u(z) ≤ Nκ v(z)+cn Nκ w(z) holds for all z ∈ ∂Rn+ . In light of (1.15) this ultimately   implies Nκ u ∈ L p (Rn−1 ). We are now ready to present the proof of Theorem 1.1. Proof of Theorem 1.1 The fact that (1.20) is included in (1.19) is an immediate consequence of Proposition 2.5. Hence, we will focus on the opposite inclusion. To this end, pick  n u ∈ C ∞ (Rn+ ) such that  u − 2∇div u = 0 in Rn+ , κ−n.t. u∂Rn exists Ln−1 − a.e. in Rn−1 , and Nκ u ∈ L p (Rn−1 ).

(3.20)

+

Let ε > 0 be arbitrary and define  n uε := u(· + εen ) ∈ C ∞ (Rn+ ) .

(3.21)

By the Lebesgue Dominated Convergence Theorem, it follows that κ−n.t.   n uε ∂Rn −→ u∂Rn =: ( f 1 , f 2 , . . . , f n ) in L p (Rn−1 ) as ε → 0+ . (3.22) +

+

In addition, (3.20) and (3.21) imply  u ε − 2∇div uε = 0 in Rn+ , Nκ uε ∈ L p (Rn−1 ),

(3.23)

and (after applying div to the identity in (3.23))    div uε = 0 in Rn+ .

(3.24)

If  κ is as in (2.39), then (1.15) and interior estimates yield   Nκ ∇ uε ∈ L p (Rn−1 ).

(3.25)

The membership in (3.25) may be obtained via a reasoning similar to that used in (2.41–2.42), this time the first inequality in (2.41) being justified by invoking [6, Theorem 11.12, p. 415]. Calderón’s version of Fatou’s theorem (cf. [3]) implies that the function div uε , which is harmonic in Rn+ and whose nontangential maximal function belongs to L p (Rn−1 ), has a nontangential trace to the boundary of Rn+ at Ln−1 -a.e. point in Rn−1 , and κ−n.t. (div uε )∂Rn ∈ L p (Rn−1 ). +

This allows us to consider the function  κ−n.t.  u ε ∂Rn in Rn+ . wε := 2Smod div +

270

(3.26)

(3.27) Reprinted from the journal

Failure of Fredholm solvability...

The properties of the operator Smod recorded in (2.21–2.23) imply ⎧ wε ∈ C ∞(Rn+ ), wε = 0 in Rn+ , ⎪ ⎪ ⎪ ⎪ ⎨ Nκ ∇wε ∈ L p (Rn−1 ), κ−n.t. (∇wε )∂Rn exists Ln−1 − a.e. in Rn−1 , ⎪ + ⎪ ⎪ κ−n.t. κ−n.t. ⎪ ⎩ ∂n wε  n = div uε  n . ∂R ∂R +

(3.28)

+

Consequently, ∂n wε is a solution of the L p -Dirichlet problem for the Laplacian in Rn+  κ−n.t. with boundary datum div u ε  n . Uniqueness for the latter problem (cf., e.g., [5]) ∂R+

implies

∂n wε = div uε in Rn+ .

(3.29)

In addition, interior estimates for wε (as in the computation in (3.13), written with wε in place of w) imply     xn (∇ 2 wε )(x) ≤ cn Nκ ∇wε (z), ∀ z ∈ ∂Rn , ∀ x = (x , xn ) ∈ κ (z). (3.30) + Next, define vε (x) := uε (x) − xn ∇wε (x), ∀ x ∈ Rn+ .

(3.31)

From this definition, (3.21), (3.28), and (3.29) first we see that κ−n.t.   n vε ∈ C ∞ (Rn+ ) , vε ∂Rn = uε ∂Rn , and + + div vε = div uε − ∂n wε = 0 in Rn+ .

(3.32)

Second, using (3.29) and (3.23) we compute  vε =  u ε − 2∇(∂n wε ) − xn ∇(wε ) =  u ε − 2∇div uε = 0 in Rn+ .

(3.33)

Third, (3.31) and (3.30) imply       Nκ ∇ vε (z) ≤ Nκ ∇ uε (z) + cn Nκ ∇wε (z), ∀ z ∈ ∂Rn+ .

(3.34)

From (3.34), (3.25), the second line in (3.28), and (1.15) we conclude   Nκ ∇ vε ∈ L p (Rn−1 ).

(3.35)

Also, having fixed an arbitrary truncation parameter ρ ∈ (0, ∞), a combination of (3.31), the second line in (3.28), and the membership in (3.23) gives Nκρ vε ∈ L p (Rn−1 ). Reprinted from the journal

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(3.36)

D. Mitrea et al.

In particular, we may apply Lemma 2.2 to obtain κ−n.t.  p n vε ∂Rn ∈ L 1 (Rn−1 ) and + κ−n.t.  κ−n.t.  ∂k vε ∂Rn = (∂k vε )∂Rn for all k ∈ {1, . . . , n − 1}. +

(3.37)

+

To continue fix k ∈ {1, . . . , n − 1}. Then (3.32), (3.33), and (3.35) imply  n ∂k vε ∈ C ∞ (Rn+ ) , (∂k vε ) = 0 and div (∂k vε ) = 0 in Rn+ , κ−n.t. (3.38) (∂k vε )∂Rn exists Ln−1 − a.e. in Rn−1 , and Nκ (∂k vε ) ∈ L p (Rn−1 ). +

These allow us to apply Proposition 2.5 and write  n−1 n−1    κ−n.t.  κ−n.t.    κ−n.t. ∂k vε n ∂Rn = − vε ) j ∂Rn . R j (∂k vε ) j ∂Rn = −∂k R j ( +

+

j=1

+

j=1

(3.39)

In concert, (3.37) and (3.39) yield  ∂k

 n−1  κ−n.t.    κ−n.t. vε n ∂Rn = −∂k vε ) j ∂Rn . R j ( +

+

j=1

(3.40)

From this and the equality in the first line of (3.32) we then see that the functions      uε n ∂Rn and − n−1 u ε ) j ∂Rn differ by a constant. Since these functions j=1 R j ( +

belong to L p (Rn−1 ), we should have

+

n−1        uε n ∂Rn = − u ε ) j ∂Rn in Rn−1 . R j ( +

+

j=1

(3.41)

Recalling now (3.22) and the fact that the Riesz  transforms are continuous on the space L p (Rn−1 ), from (3.41) we obtain f n = − n−1 j=1 R j f j , as wanted.

Compliance with ethical standards Conflicts of interest The authors declare that they have no conflict of interest.

References 1. Bitsadze, A.V.: Ob edinstvennosti resheniya zadachi Dirichlet dlya ellipticheskikh uravnenii s chastnymi proizvodnymi. Uspekhi Matem. Nauk 3(6), 211–212 (1948) 2. Bitsadze, A.V.: Boundary Value Problems for Second-Order Elliptic Equations. North-Holland, Amsterdam (1968) 3. Calderón, A.P.: On the behavior of harmonic functions near the boundary. Trans. Am. Math. Soc. 68, 47–54 (1950)

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Failure of Fredholm solvability... 4. Marín, J.J., María Martell, J., Mitrea, D., Mitrea, I., Mitrea, M.: Singular Integrals, Quantitative Flatness, and Boundary Problems, Book Manuscript (2020) 5. Martell, J.M., Mitrea, D., Mitrea, I., Mitrea, M.: The Dirichlet problem for elliptic systems with data in Köthe function spaces. Rev. Mat. Iberoam. 32(3), 913–970 (2016) 6. Mitrea, D.: Distributions, Partial Differential Equations, and Harmonic Analysis, 2nd edn. Springer, Cham (2018) 7. Mitrea, D., Mitrea, I., Mitrea, M.: A sharp divergence theorem with nontangential traces. Notices AMS 67(9), 1295–1305 (2020) 8. Mitrea, D., Mitrea, I., Mitrea, M.: Geometric Harmonic Analysis. Book Manuscript (2020) 9. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton Mathematical Series, Vol. 43, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton (1993) Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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273

Analysis and Mathematical Physics (2020) 10:46 https://doi.org/10.1007/s13324-020-00392-1

Local regularity of axisymmetric solutions to the Navier–Stokes equations G. Seregin1,2 Received: 24 June 2020 / Revised: 4 August 2020 / Accepted: 4 September 2020 / Published online: 10 September 2020 © Springer Nature Switzerland AG 2020

Abstract In the note, a local regularity condition for axisymmetric solutions to the non-stationary 3D Navier–Stokes equations is proven. It reads that axially symmetric energy solutions to the Navier–Stokes equations have no Type I blowups. Keywords Navier–Stokes equations · Axisymmetric solutions · Local regularity

1 Introduction The aim of the note is to discuss potential singularities of axisymmetric solutions to the non-stationary 3D Navier–Stokes equations. Roughly speaking, we would like to show that if scale-invariant energy quantities of an axially symmetric solution are bounded then such a solution is smooth. By definition, potential singularities with bounded scale-invariant energy quantities are called Type I blowups. It is important to notice that our result does not follow from the so-called ε-regularity theory, where regularity is coming from smallness of those scale-invariant energy quantities. Before stating and proving the main result of the note, see Theorem 2.1, we are going to remind basic notions from the mathematical theory of the Navier–Stokes equations. For simplicity, let us consider the following Cauchy problem for the Navier–Stokes equations: div v = 0 (1.1) ∂t v + v · ∇v − v = −∇q, in Q + = R3 ×]0, ∞[ and

v|t=0 = u 0

(1.2)

Dedicated to Vladimir Gilelevich Mazya.

B

G. Seregin [email protected]

1

OxPDE, Mathematical Institute, University of Oxford, Oxford, UK

2

St Petersburg Department of V A Steklov Mathematical Institute, Saint Petersburg, Russia

Chapter 13 was originally published as Seregin, G. Analysis and Mathematical Physics (2020) 10:46. https://doi.org/10.1007/s13324-020-00392-1. Reprinted from the journal

275

G. Seregin ∞ (R3 ) := {v ∈ C ∞ (R3 ) : div v = 0}. in R3 , where u 0 ∈ C0,0 0 One of the main problems of the mathematical theory of viscous incompressible fluids is the global well-posedness of the Cauchy problem (1.1) and (1.2). A plausible approach (which, of course, is not unique) is to prove the global existence of a solution and then to prove its uniqueness. It was done by J. Leray many years ago in [10], who introduced the notion which is known now as a weak Leray–Hopf solution.

Definition 1.1 A divergence free velocity field v is a weak Leray–Hopf solution to the Cauchy problem (1.1) and (1.2) if it has the following properties: ∞ (R3 ) in L (R3 ) 1. v ∈ L ∞ (0, ∞; J ) ∩ L 2 (0, ∞; J21 ), where J is the closure of C0,0 2  1 2  ∞ (R3 ) with respect to the semi-norm and J21 the closure of C0,0 |∇w|2 d x ; R3  2. the function t → v(x, t) · w(x)d x is continuous on [0, ∞[ for any w ∈ L 2 (R3 ); R3

3. the Navier–Stokes equations is satisfied as the variational identity  

 v · ∂t w + v ⊗ v : ∇w − ∇v : ∇w d xdt = 0

Q+

for any test vector-valued function w ∈ C0∞ (Q + ) with div w = 0; 4. v(·, t) − u 0 (·) L 2 (R3 ) → 0 as t ↓ 0; 5. global energy inequality 1 2

t 

 |v(x, t)| d x + 2

R3

1 |∇v| d xdt ≤ 2 2



0 R3

 |u 0 |2 d x R3

holds for all t ≥ 0. There is no information about the pressure in Definition 1.1. But one can easily recover the pressure by means of the linear theory and the following estimate for the pressure takes place q L 3 (Q + ) ≤ c v 2L 3 (Q + ) . 2

There is an important class of weak solutions, which is closely related to the uniqueness of solutions to the initial boundary value problems for the Navier–Stokes equations. Definition 1.2 Let v be a weak Leray–Hopf solution to (1.1) and (1.2). It is a strong solution to the Cauchy problem on the set Q T = R3 ×]0, T [ if ∇v ∈ L 2,∞ (Q T ). The proposition, proved by Leray in [10], essentially reads the following. Assume that v a weak Leray–Hopf solution to (1.1) and (1.2). Then there exists a number 276

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Local regularity of axisymmetric solutions…

T ≥ c ∇u 0 −4 , where c is an universal constant, such that v is strong solution in L 2 (R3 ) QT . Another important result proved by Leray is the so-called weak-strong uniqueness. Assume that v 1 is another weak Leray–Hopf solution with the same initial data u 0 , then v 1 = v on Q T . So, as it follows from the above statement, in order to prove uniqueness of weak Leray–Hopf solution on the interval ]0, T [, it is enough to show that ∇v ∈ L 2,∞ (Q T ). In other words, the problem of unique solvability of the Cauchy problem in the energy class can be reduced to the problem of regularity of weak Leray–Hopf solutions. As usual in the theory of non-linear equations, we do not need to prove that ∇v ∈ L 2,∞ (Q T ) for some T > 0. In fact, it is enough to prove a weaker regularity result and then the remaining part of the proof of regularity follows from the linear theory. For example, one of the convenient spaces for such intermediate regularity is the space L ∞ (Q T ). So, the first time when a singularity occurs can be defined as follows: lim sup v(·, t) L ∞ (R3 ) = ∞. t↑T

To study regularity of weak Leray–Hopf solutions by classical PDE’s methods, we should mimic energy solutions on the local (in space-time) level. To this end, the pressure should be involved into considerations. The corresponding setting has been already discussed by Caffarelli–Kohn–Nirenberg, who have introduced the notion of suitable weak solutions to the Navier–Stokes equations, see [1,11,14,15]. Definition 1.3 Let ω ⊂ R3 and T2 > T1 . w and r is a suitable weak solution to the Navier–Stokes in Q ∗ = ω×]T1 , T2 [ if: 1. w ∈ L 2,∞ (Q ∗ ), ∇w ∈ L 2 (Q ∗ ), r ∈ L 3 (Q ∗ ); 2 2. w and r satisfy the Navier–Stokes equations in the sense of distributions; 3. for a.a. t ∈ [T1 , T2 ], the local energy inequality t 

 ϕ(x, t)|w(x, t)| d x + 2 ω



t 

ϕ|∇w| d xdt ≤

2

2

T1 ω

[|w|2 (∂t ϕ + ϕ) T1 ω 

+ w · ∇ϕ(|w| + 2r )]d xdt 2

holds for all non-negative ϕ ∈ C01 (ω×]T1 , T2 + (T2 − T1 )/2[). In order to state a typical result of the regularity theory for suitable weak solutions, introduce the notation for parabolic cylinders (balls): Q(z 0 , R) = B(x0 , R)×]t0 − R 2 , t0 [, B(x0 , R) = {x ∈ R3 : |x − x0 | < R}, and z 0 = (x0 , t0 ). Proposition 1.4 1. There are universal constants ε and c0 such that for any suitable weak solution v and q in Q(z 0 , R) satisfying the assumption C(z 0 , R) + D(z 0 , R) < ε, Reprinted from the journal

277

G. Seregin

where 1 C(z 0 , R) = 2 R



1 D(z 0 , R) = 2 R

|v| dz, 3

Q(z 0 ,R)



3

|q| 2 dz, Q(z 0 ,R)

the velocity field v is Hölder continuos in Q(z 0 , R/2) and sup

z Q(z 0 ,R/2)

|v(z)| ≤

c0 . R

2. There is a universal constant ε > 0 such that for any suitable weak solution v and q in Q(z 0 , R) satisfying the assumption g(z 0 ) := min{lim sup E(z 0 , r ), lim sup A(z 0 , r ), lim sup C(z 0 , r )} < ε, r →0

r →0

r →0

where A(z 0 , R) =

sup t0

−R 2 0, set σ N (x, t) = σ (x, t) if |σ (x, t)| ≤ N , σ N (x, t) = N if σ > N , and σ N (x, t) = −N if σ < −N . Take two cut-off functions: the first of them is ψ = ψ(x, t), vanishing in a neighbourhood of the parabolic boundary of Q and the second one is φ = φ(x  ), vanishing in a neighbourhood of the axis of symmetry, multiple the left hand side of equation (2.2) by σ N2m−1 ψ 4 φ 2 and integrate by parts. As a result, three different terms appear and they will be treated 280

Reprinted from the journal

Local regularity of axisymmetric solutions…

separately. For the first one, we have t∗  ∂t σ σ N2m−1 ψ 4 φ 2 dz = −1 C

1 2m



 σ N2m ψ 4 φ 2 |(x,t∗ ) d x C

(σ − σ N )σ N2m−1 ψ 4 φ 2 |(x,t∗ ) d x

+ C

1 − 2m

t∗ 

t∗  σ N2m ∂t ψ 4 φ 2 dz

(σ − σ N )σ N2m−1 ∂t ψ 4 φ 2 dz.

−

−1 C

−1 C

Since the second term on the right hand side of the above identity is non-negative, it can be dropped out: t∗ 

1 ≥ 2m

∂t σ σ N2m−1 ψ 4 φ 2 dz −1 C

1 − 2m

 σ N2m ψ 4 φ 2 |(x,t∗ ) d x C

t∗ 

t∗  σ N2m ∂t ψ 4 φ 2 dz

−1 C

(σ − σ N )σ N2m−1 ∂t ψ 4 φ 2 dz.

− −1 C

Denoting b = 2(x  , 0)|x  |−2 , transform the second term as follows: t∗  (v + b) · ∇σ σ N2m−1 ψ 4 φ 2 dz −1 C

1 =− 2m

t∗  (v + b) · ∇(ψ 4 φ 2 )σ N2m dz −1 C

t∗  −

(v + b) · ∇(ψ 4 φ 2 )(σ − σ N )σ N2m−1 dz

−1 C

Finally, for the third term, we have t∗  −

σ σ N2m−1 ψ 4 φ 2 dz

−1 C

2m − 1 = m2

Reprinted from the journal

t∗  |∇σ Nm |2 ψ 4 φ 2 dz −1 C

281

1 − 2m

t∗  σ N2m (ψ 4 φ 2 )dz −1 C

G. Seregin

t∗  (σ − σ N ) · σ N2m−1 (ψ 4 φ 2 )dz.

− −1 C

Combining previous relationships, we find the following energy inequality 

1 2m

σ N2m ψ 4 φ 2 |(x,t∗ ) d x C

≤

2m − 1 + m2

t∗  |∇σ Nm |2 ψ 4 φ 2 dz −1 C

t∗ 

1 2m

σ N2m (∂t (ψ 4 φ 2 ) + (ψ 4 φ 2 ))dz

−1 C t ∗  

(σ − σ N )σ N2m−1 (∂t (ψ 4 φ 2 ) + (ψ 4 φ 2 ))dz

+ −1 C

1 + 2m

t∗  (v + b) · ∇(ψ 4 φ 2 )σ N2m dz

−1 C t ∗  

(v + b) · ∇(ψ 4 φ 2 )(σ − σ N )σ N2m−1 dz.

+ −1 C

Now, selecting a special non-negative cut-off function φ so that ψ(x  ) = 0 if 0 < |x  | < ε/2, ψ(x  ) = 1 if |x  | > ε, and |∇ k φ| ≤ cε−k , k = 0, 1, 2, let us see what happens if ε → 0. We start with the two most important terms: t∗  I1 =

|v|ψ 4 φ|∇φ|(|σ | + |σ N |)|σ N |2m−1 dz −1 C

and t∗  I2 =

|b|ψ 4 φ|∇φ|(|σ | + |σ N |)|σ N |2m−1 dz. −1 C

As to I1 , it is easy to see t∗ I1 ≤ c −1

1

 |v|2 N 2m−1 d d x3 dt → 0

2π −1 ε/2