S2-type parametric surfaces with prescribed mean curvature and minimal energy

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S2 -TYPE PARAMETRIC SURFACES WITH PRESCRIBED MEAN CURVATURE AND MINIMAL ENERGY∗

Paolo CALDIROLI Dipartimento di Matematica Universit` a di Torino via Carlo Alberto, 10 – 10123 Torino, Italy E-mail: [email protected] Roberta MUSINA Dipartimento di Matematica ed Informatica Universit` a di Udine via delle Scienze, 206 – 33100 Udine, Italy E-mail: [email protected]

Abstract. Given a function H ∈ C 1 (R3 ) asymptotic to a constant at infinity, we investigate the existence of nontrivial, conformal surfaces parametrized by the sphere, with mean curvature H and minimal energy.

1. Introduction In this paper we deal with two-dimensional parametric surfaces in R3 of prescribed mean curvature. In particular we are interested in the existence of surfaces parametrized by the sphere S2 in case the prescribed mean curvature is nonconstant. This problem is motivated both for its geometrical interest, and by the fact that it arises as a mathematical model in capillarity theory to describe interface surfaces in presence of external forces (see [4], [6]). The problem admits an analytical formulation, as follows: given a smooth function H: R3 → R, find a nonconstant, conformal function ω: R2 → R3 , smooth as a map on S2 , satisfying:  in R2 x ∧ ωy R∆ω = 2H(ω)ω (1.1) 2 |∇ω| < +∞ . R2

∗ Work

supported by M.U.R.S.T. progetto di ricerca “Metodi Variazionali ed Equazioni Differenziali Nonlineari” (cofin. 2001/2002)

1

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P. Caldiroli & R. Musina

Here subscripts mean partial differenziation, the symbols ∆ and ∇ represent respectively the Laplacian (in R2 ) and the gradient, and ∧ denotes the exterior product in R3 . A solution ω of (1.1) satisfying the above listed requirements will be called shortly an H-bubble. Note that problem (1.1) is invariant with respect to the conformal group. This means that we deal with a problem on the image of ω, rather that on the mapping ω itself. When the prescribed mean curvature is a nonzero constant H(u) ≡ H0 , Brezis and Coron [2] proved that the only nonconstant solutions to (1.1) are spheres of radius |H0 |−1 anywhere placed in R3 . Only recently, the case in which H is nonconstant has been investigated (see [3]). Here we are interested in a class of curvature functions which are asymptotic to a constant at infinity. This case has been already considered in [3], but in the present work we follow a different strategy that allows us to carry out some improvements, to get more general results and to exhibit new features in the problem. The first important remark on problem (1.1) concerns its variational nature. That is, solutions to (1.1) can be detected as critical points of a suitable energy functional. More precisely, the energy functional associated to problem (1.1) is defined by Z Z 1 2 |∇u| + 2 QH (u) · ux ∧ uy , EH (u) = 2 R2 R2 where QH : R3 → R3 is any vector field such that div QH = H. For several reasons it is meaningful not only to look for H-bubbles, but also to study the existence of H-bubbles with minimal energy. Hence, we are lead to consider the following problems: 1) calling BH the set of H-bubbles, find conditions on H ensuring that BH 6= ∅; 2) assuming that BH 6= ∅, study the minimization problem: µH = inf EH . BH

Having in mind a curvature function H which is asymptotic to a constant at infinity, we start with the simplest case in which H is assumed to be constant far out. Theorem 1.1 Let H ∈ C 1 (R3 ) satisfy (h1 ) H(u) = H∞ ∈ R \ {0} as |u| ≥ R, for some R > 0, (h2 ) supu∈R3 |H(u)u − 3QH (u)| < 12 . Then there exists ω ∈ BH such that EH (ω) = µH . Moreover µH ≤

4π 2 . 3H∞

The assumption (h2 ), roughly speaking, measures how far H differs from the constant value H∞ . In principle, it depends on the choice of the vector field QH which is not uniquely defined. In fact, we can replace the hypothesis (h2 ) with a weaker, but less explicit condition which actually depends just on H or, more

S2 -type parametric surfaces

3

precisely, on the radial component of ∇H (see Section 3). Since, by (h1 ) one immediately has that BH is nonempty, the problem reduces to investigate the semicontinuity of the energy functional along a sequence of H-bubbles. As shown by Wente [10], in general EH is not globally semicontinuous, even if H is constant. However, along a sequence of solutions of (1.1), thanks to the condition (h2 ), semicontinuity holds true. As a next step we want to give up the condition (h1 ), by considering the more general case of a prescribed mean curvature which is just asymptotic to a constant at infinity. Clearly, now it is not known if BH is nonempty. The strategy consists in approximating H with a sequence of functions (Hn ) which are constant far out and for which Theorem 1.1 can be applied, and then, passing to the limit. Theorem 1.2 Let H ∈ C 1 (R3 ) satisfy (h2 ) and (h3 ) H(u) → H∞ as |u| → ∞, for some H∞ ∈ R, (h4 ) supu∈R |∇H(u) · u| < +∞. Then there exists a sequence (Hn ) ⊂ C 1 (R3 ) such that Hn → H uniformly on R3 and, for every n ∈ N, BHn is nonempty and inf BHn EHn = µHn is attained. If, in addition (h5 ) lim inf µHn
0 u6=0

(1.2)

which represents the mountain pass level along radial paths. Now, the existence of minimal H-bubbles can be stated as follows. Theorem 1.3 (see [3]) Let H ∈ C 1 (R3 ) satisfy (h3 ), (h6 ) supu∈R3 |∇H(u) · u u| < 1,

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(h7 ) cH
H∞ on a suitably large set. Moreover, the condition (h6 ) allows us to find also a lower bound for the minimal energy µH and precisely µH ≥

4π . 3kHk2∞

We finally remark that all the previous Theorems give no information about the position of the minimal H-bubble, but only the information on its energy is available. Hence, the same results hold true if in all the statements the assuptions are fulfilled by H(· + p) for some p ∈ R3 . The work is organized as follows: in Section 2 we fix the notation and we state some preliminaries in view of a variational approach to problem (1.1). Section 3 contains the proof of a weaker version of Theorem 1.1. Moreover a semicontinuity result is discussed. Section 4 concerns the case H asymptotic to a constant at infinity. In particular we show a genaralization of Theorem 1.2 and we make some remarks about Theorem 1.3. For the complete proof of Theorem 1.3 we refer to [3]. 2. Preliminaries In this Section we introduce a variational setting suited to study problem (1.1). We note that all the statements of this Section hold true assuming just H ∈ C 1 (R3 )∩L∞ or, sometimes, even H ∈ L∞ (R3 ). Firstly, as a variational space we will take the Hilbert space of functions u: R2 → 3 R with finite Dirichlet integral, which is isomorphic to H 1 (S2 , R3 ). It can be defined as follows. Consider the mapping ω 0 : R2 → S2 given by   µx 2 , (2.1) ω 0 (z) =  µy  , µ = µ(z) = 1 + |z|2 1−µ being z = (x, y) and |z|2 = x2 + y 2 . Observe that ω 0 is the inverse of the standard stereographic projection, it is a conformal parametrization of the unit sphere (centered at 0). According to the definition given in the Introduction, ω 0 is an H-bubble, with H ≡ 1. Then, set X = {ϕ ◦ ω 0 : ϕ ∈ H 1 (S2 , R3 )} .

S2 -type parametric surfaces

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The space X naturally inherits a Hilbertian structure from H 1 (S2 , R3 ). In particular, the inner product in X can be defined as hu, vi = hu ◦ π, v ◦ πiH 1 (S2 ,R3 ) where π: S2 → R2 is the standard stereographic projection. Explicitely, one has Z
∇u · ∇v + µ2 u · v , hu, vi = R2

with µ given by (2.1). Hence the space X can be equivalently defined as Z
1 |∇u|2 + µ2 |u|2 < +∞} . X = {u ∈ Hloc (R2 , R3 ) : R2

X turns out to be a Hilbert space, endowed with the norm kuk = hu, ui1/2 . For ˆ ∈ X and kˆ uk = kuk. In every u ∈ X set u ˆ(z) = u(ˆ z ), where zˆ = |z|z 2 . Then, u particular, for every R > 0 Z Z |∇ˆ u|2 = |∇u|2 . (2.2) 1 |z|> R

|z| 0 such that |VH (u)|2/3 ≤ CH D(u) for every u ∈ X ∩ L∞ .

(2.3)

Furthermore, Steffen in [9] proves that the functional VH admits a continuous extension on X and then, (2.3) holds true also for every u ∈ X. As far as concerns the differentiability of VH , the following result is useful. This result is also known and we refer to [7] for a proof. Lemma 2.2 If H ∈ C 1 (R3 ) ∩ L∞ , then for every u ∈ X and for h ∈ Cc∞ (R2 , R3 ) the directional derivative of VH at u along h exists, and it is given by Z H(u)h · ux ∧ uy . dVH (u)h = R2

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P. Caldiroli & R. Musina

Moreover, for every u ∈ X ∩ L∞ one has Z dVH (u)u = H(u)u · ux ∧ uy . R2

Remark 2.3 By Lemma 2.2, the functional VH does not depend on the choice of the vector field QH , but only on H. Moreover, if H is constant, then VH ∈ C 1 (X, R), while if H is nonconstant, in general the functional VH is not even Gˆateauxdifferentiable at every u ∈ X. The following result will be also useful in the sequel. Lemma 2.4 If H ∈ C 1 (R3 ) is such that H(u) = 0 as |u| ≥ R, then for every u ∈ X ∩ L∞ one has VH (u) = VH (π R ◦ u) , where π R : R3 → B R is the retraction on the ball B R = {u ∈ R3 : |u| ≤ R}. R1 Proof. Take QH (u) = mH (u)u, with mH (u) = 0 H(su)s2 ds. Setting uR = π R ◦ u, one has Z
R mH uR uR · uR VH (uR ) = x ∧ uy R2 Z Z
R3 u · ux ∧ uy = mH (u)u · ux ∧ uy + mH uR |u|3 {z:|u(z)|≤R} {z:|u(z)|>R}   Z
R3 u · ux ∧ uy . = VH (u) − mH (u) − mH uR |u|3 {z:|u(z)|>R} But one can easily check that if |u| > R then
R3 mH (u) = mH π R (u) . |u|3
Hence VH uR = VH (u).

The energy functional EH : X ∩ L∞ → R is defined by EH (u) = D(u) + 2VH (u) .

By virtue of the above discussed properties for VH , the functional EH admits a continuous extension on X. Remark 2.5 The failure of lower semicontinuity of VH (and then of EH ) can be shown by the following example, essentially due to Wente [10]. Take, for simplicity, H ≡ 1, let ω 0 : R2 → S2 be given by (2.1) and set ω ǫ (z) = ω 0 (ǫz). Then for 3 every λ ∈ R one has D(λω ǫ ) = λ2 D(ω 0 ) = 4πλ2 , VH (λω ǫ ) = λ3 VH (ω 0 ) = − 4π 3 λ . However ω ǫ → ω 0 (0) = e3 weakly in X, and E(λe3 ) = 0. Thanks to Lemma 2.2, critical points of EH correspond to weak solutions to problem (1.1). In the following statement we collect some facts about (weak) solutions to (1.1).

S2 -type parametric surfaces

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Lemma 2.6 Let H ∈ C 1 (R3 ) ∩ L∞ . If ω ∈ X is a weak solution to (1.1), i.e., dEH (ω)h = 0 for every h ∈ Cc∞ (R2 , R3 ), then:

(i) ω ∈ C 3 (R2 , R3 ) is a classical solution to (1.1), (ii) ω is conformal, and smooth as a map on S2 ,

(iii) diam ω ≤ C(1 + D(ω)) where C is a constant depending only on kHk∞ . All the above statements are known in the literature and we refer to [5] for a proof (see also [1] Theorem 4.10). 3. The case H constant far out The goal of this Section is to prove Theorem 1.1. Actually, the main result here is a semicontinuity property for the energy functional along a sequence of critical points (Theorem 3.1). This result can be stated in a more general form that will be useful in the sequel and that allows us to recover a weaker version of Theorem 1.1 (see Theorem 3.3). Firstly, given H ∈ C 1 (R3 ) ∩ L∞ , let JH (u) = ∇H(u) · u and assume (h4 ), that is, JH ∈ L∞ (R3 ). Let us introduce the functional WH : X ∩ L∞ → R defined by Z (3QH (u) − H(u)u) · ux ∧ uy . WH (u) = R2

According to Remark 2.3, since div(H(u)u − 3QH (u)) = JH (u), the functional WH depends just on JH , and not on QH . In particular, WH = WH ′ if H and H ′ differ by an additive constant. Let us set D(u) . λH = inf ∞ u∈X∩L |WH (u)| It is also convenient to introduce for every R > 0 the value λH,R =

inf

u∈X kuk∞ ≤R

D(u) . |WH (u)|

Clearly, the mapping R 7→ λH,R is decreasing and λH,R → λH as R → +∞. Since WH (u) = 3VH (u) − dVH (u)u on X ∩ L∞ , if u ∈ X and kuk∞ ≤ R, then   2 D(u) ≤ 3EH (u) − dEH (u)u . (3.1) 1− λH,R When λH > 2 (or λH,R > 2), (3.1) is useful to get a bound of the Dirichlet norm in terms of the energy. Note that, if νH = 2 supu∈R3 |H(u)u − 3QH (u)|, then λH ≥ 1/νH . In particular, (h2 ) implies λH > 2. The role of λH (or λH,R ) is made clear by the following fact, which states a semicontinuity property along a sequence of solutions.

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Theorem 3.1 Let (Hn ) ⊂ C 1 (R3 ), H ∈ C 1 (R3 ) and R > 0 be such that: (i) Hn → H uniformly on B R = {u ∈ R3 : |u| ≤ R}, (ii) λHn ,R ≥ 2 for every n ∈ N, (iii) for every n ∈ N there exists an Hn -bubble ω n with kω n k∞ ≤ R, |∇ω n (0)| = k∇ω n k∞ = 1, and supn k∇ω n k2 < +∞. Then there exists an H-bubble ω such that, for a subsequence, ω n → ω weakly in X 1 and strongly in Cloc (R2 , R3 ). Moreover EH (ω) ≤ lim inf EHn (ω n ) . Proof. We split the proof into some steps. Step 1. There exists ω ∈ X ∩ C 1 (R2 , R3 ) such that, for a subsequence, ω n → ω 1 weakly in X and strongly in Cloc (R2 , R3 ). From the assumption (iii), there exists ω ∈ X such that, for a subsequence, ω n → ω weakly in X. Now, we show that for every ρ > 0 and for every p > 1 the sequence (ω n ) is bounded in H 2,p (Dρ , R3 ). To this extent, we will use the following regularity estimate (which is a special case of Lemma A.4 in [3]). Lemma 3.2 Let H ∈ C 1 (R3 ) ∩ L∞ . Then there exists ε = ε(kHk∞ ) > 0 and, for every p ∈ (1, +∞) a constant Cp = Cp (kHk∞ ) > 0, such that if u: Ω → R3 is a weak solution to ∆u = 2H(u)ux ∧ uy in Ω (with Ω open domain in R2 ), then 2

k∇ukL2 (DR (z)) ≤ ε ⇒ k∇ukH 1,p (DR/2 (z)) ≤ Cp R p −2 k∇ukL2 (DR (z)) for every disc DR (z) ⊂ Ω, with R ∈ (0, 1]. Since by (iii) we are interested in the convergence of a sequence in the region B R , using (i), we may assume that Hn → H uniformly on R3 . Hence, by Lemma 3.2, for every n ∈ N there exists εn > 0 and Cp,n > 0 for which 2

k∇ω n kL2 (DR (z)) ≤ εn ⇒ k∇ω n kH 1,p (DR/2 (z)) ≤ Cp,n R p −2 k∇ω n kL2 (DR (z)) for every z ∈ R2 and for every R ∈ (0, 1]. By (i), one has εn ≥ ε > 0 and Cp,n ≤ Cp for every n ∈ N. Fix ρ > 0. Since k∇ω n k∞ = 1, there exists R > 0 and a finite covering {DR/2 (zi )}i∈I of Dρ such that k∇ω n kL2 (DR (zi )) ≤ ε for every n ∈ N and i ∈ I. Since kω n k∞ ≤ R, we have that kω n kH 2,p (DR/2 (zi )) ≤ C¯p,R for some constant C¯p,R > 0 independent of i ∈ I and n ∈ N. Then the sequence (ω n ) is bounded in H 2,p (Dρ , R3 ). For p > 2 the space H 2,p (Dρ , R3 ) is compactly embedded into C 1 (Dρ , R3 ). Hence ω n → ω strongly in C 1 (Dρ , R3 ). By a standard diagonal 1 (R2 , R3 ). argument, one concludes that ω n → ω strongly in Cloc Step 2. ω is an H-bubble.

For every n ∈ N one has that if h ∈ Cc∞ (R2 , R3 ) then Z Z n Hn (ω n )h · ωxn ∧ ωyn = 0 . ∇ω · ∇h + 2 R2

R2

S2 -type parametric surfaces

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By Step 1, passing to the limit, one immediately infers that ω is a weak solution to (1.1). According to Lemma 2.6, ω is a classical, conformal solution to (1.1). In addition, ω is nonconstant, since |∇ω(0)| = lim |∇ω n (0)| = 1. Hence ω is an H-bubble. Step 3. EH (ω) ≤ lim inf EHn (ω n ). By Step 1, for every r > 0, one has EHn (ω n , Dr ) → EH (ω, Dr )

(3.2)

where we denote EHn (u, Ω) =

1 2

Z

Ω

|∇u|2 + 2

Z

Ω

QHn (u)u · ux ∧ uy

(and similarly for EH (u, Ω)). Now, fixing ǫ > 0, let r > 0 be such that E (ω, R2 \ Dr ) ≤ ǫ ZH |∇ω|2 ≤ ǫ .

(3.3) (3.4)

R2 \Dr

By (3.3) and (3.2) we have EH (ω) ≤

EH (ω, Dr ) + ǫ

= EHn (ω n , Dr ) + ǫ + o(1)

= EHn (ω n ) − EHn (ω n , R2 \ Dr ) + ǫ + o(1)

(3.5)

with o(1) → 0 as n → +∞. Now, by (iii) we have Z

∂Dr

ωn ·

∂ω n ∂ν

Z


|∇ω n |2 + ω n · ∆ω n R2 \Dr Z Z Hn (ω n )ω n · ωxn ∧ ωyn |∇ω n |2 + 2 = R2 \Dr R2 \Dr Z 1 = 3EHn (ω n , R2 \ Dr ) − |∇ω n |2 2 R2 \Dr Z Gn (ω n ) · ωxn ∧ ωyn , −2 =

(3.6)

R2 \Dr

where we set Gn (u) = 3QHn (u) − Hn (u)u. Hence (3.5)–(3.6) give EH (ω) ≤

Z Z 1 1 ∂ω n EHn (ω n ) − − ωn · |∇ω n |2 3 ∂Dr ∂ν 6 R2 \Dr Z 2 Gn (ω n ) · ωxn ∧ ωyn + ǫ + o(1) . − 3 R2 \Dr

(3.7)

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1 Since ω n → ω strongly in Cloc (R2 , R3 ) and since ω is an H-bubble, we have Z Z n ∂ω n ∂ω ω· = lim ω · n→+∞ ∂D ∂ν ∂ν ∂Dr r Z
2 ω · ∆ω + |∇ω| = R2 \Dr Z
2 2H(ω)ω · ωx ∧ ωy + |∇ω| = R2 \Dr Z
≤ kωk∞ kHkL∞ (BR ) + 1 |∇ω|2 R2 \Dr


kωk∞ kHkL∞ (BR ) + 1 ǫ

≤

(3.8)

because of (3.4). Now we estimate Z Z 1 2 n 2 − |∇ω | − Gn (ω n ) · ωxn ∧ ωyn . 6 R2 \Dr 3 R2 \Dr For every n ∈ N, let hn ∈ H 1 (Dr , R3 ) be the harmonic extension of ω n |∂Dr , and set  n h (z) as |z| < r n u (z) = ω n (z) as |z| ≥ r . Note that un ∈ X, and kun k∞ ≤ R, because kω n k∞ ≤ R and khn kL∞ (Dr ) ≤ kω n kL∞ (∂Dr ) . Therefore, using (ii) we obtain λHn ,R |WHn (un )| ≤ D(un ) and then Z Z 1 2 n 2 − |∇ω | − Gn (ω n ) · ωxn ∧ ωyn 6 R2 \Dr 3 R2 \Dr 2 1 = − D(un ) − WHn (un ) 3Z 3 Z 1 2 n 2 + |∇h | + Gn (hn ) · hnx ∧ hny 6 Dr 3 Dr Z Z 2 1 n 2 |∇h | + Gn (hn ) · hnx ∧ hny ≤ 6 Dr 3 Dr  Z 1 2 ≤ + kHn kL∞ (BR ) R |∇hn |2 . (3.9) 6 3 Dr Since, by Step 1, ω n ∂D → ω ∂D in C 1 , one gets r r Z Z |∇hn |2 → |∇h|2 Dr

Dr

where h ∈ H 1 (Dr , R3 ) is the harmonic extension of ω|∂Dr . Setting v(z) = ω as |z| ≤ r, one has that v ∂D = ω ∂D and by (2.2) r r Z Z Z |∇h|2 ≤ |∇v|2 = |∇ω|2 . Dr

Dr

R2 \Dr



r2 z |z|2



S2 -type parametric surfaces

Therefore, by (3.4) and (3.9) we obtain that Z Z 1 2 − |∇ω n |2 − Gn (ω n ) · ωxn ∧ ωyn ≤ Cǫ + o(1) . 6 R2 \Dr 3 R2 \Dr

11

(3.10)

In conclusion, (3.7), (3.8), and (3.10) yield EH (ω) ≤ EHn (ωn ) + Cǫ + o(1) with o(1) → 0 as n → +∞. By the arbitrariness of ǫ > 0, the thesis follows. As a consequence of Theorem 3.1 we infer the following result which, in fact, is a generalized version of Theorem 1.1. Theorem 3.3 Let H ∈ C 1 (R3 ) satisfy (h1 ) and λH > 2. Then there exists ω ∈ BH 4π such that EH (ω) = µH . Moreover µH ≤ 3H 2 . ∞ Proof. The assumption (h1 ) guarantees that BH is nonempty, since the spheres of radius |H∞ |−1 placed in the region |u| ≥ R are H-bubbles. In particular, this 4π n n implies that µH ≤ 3H 2 . Now, take a sequence (ω ) ⊂ BH with EH (ω ) → µH . ∞ Since the problem (1.1) is invariant with respect to the conformal group, we may assume that k∇ω n k∞ = |∇ω n (0)| = 1. Since λH > 2, (3.1) yields that sup k∇ω n k2 < +∞ .

(3.11)

sup kω n k∞ < +∞ .

(3.12)

We may also assume that Indeed, by Lemma 2.6 and by (3.11), there exists ρ > 0 such that diam ω n ≤ ρ for every n ∈ N. If kω n k∞ ≤ R + ρ, set ω ˜ n = ω n . If kω n k∞ > R + ρ, then n n by the assumption (h1 ), ω solves ∆u = 2H∞ u x ∧ uy . Let pn ∈ range ω be such that |pn | = kω n k∞ . Set qn =

1−

R+ρ |pn |

pn and ω ˜ n = ω n − qn . Then

k˜ ω n k∞ ≤ R + ρ, and |˜ ω n (z)| ≥ R for every z ∈ R2 . Hence, also ω ˜ n ∈ BH , n n n n ω ) = EH (ω ). The new sequence (˜ ω ) satisfies (3.11), (3.12) and EH (˜ ω ) = EH∞ (˜ and EH (˜ ω n ) → µH . Hence, we are in position to apply Theorem 3.1 (with Hn = H for every n ∈ N) and then the conclusion follows. 4. The case H asymptotic to a constant at infinity

In this Section, we will prove the following result. Theorem 4.1 Let H ∈ C 1 (R3 ) satisfy (h3 ), (h4 ) and λH > 2. Then there exists a sequence (Hn ) ⊂ C 1 (R3 ) converging to H uniformly on R3 and such that for every n ∈ N there exists ω n ∈ BHn with EHn (ω n ) = µHn . If, in addition, (h5 ) holds, then there exists ω ∈ BH with EH (ω) = µH . Since the condition λH > 2 implies (h2 ), Theorem 1.2 follows as a special case of the above result. The first step in order to prove Theorem 4.1 is given by the following result.

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Lemma 4.2 Let H ∈ C 1 (R3 ) satisfy (h4 ) and H(u) → 0 as |u| → +∞. Then there exists a sequence (Hn ) ⊂ Cc1 (R3 ) such that uniformly on R3 ,

Hn → H

λHn → λH .

(4.1) (4.2)

Proof. Let (Rn ) and (δn ) be two sequences in (0, +∞) such that Rn → +∞ and δn → 0. For every n ∈ N let χn ∈ C 1 (R, [0, 1]) satisfy χn (r) = 1 as r ≤ Rn , χn (r) = 0 as r ≥ Rn + δn , and |χ′n (r)| ≤ 2δn−1 . Let J, Jn : R3 → RR be defined by +∞ J(u) = ∇H(u)·u and Jn (u) = χn (|u|)J(u). Noting that H(u) = − 1 J(su)s−1 ds 3 (for u 6= 0), let us define for u ∈ R \ {0} Z +∞ Hn (u) = − Jn (su)s−1 ds . 1

We will see below that actually Hn can be extended continuously at u = 0. Step 1. Hn ∈ Cc1 (R3 ) and Hn → H uniformly on R3 . We have

Hn (u)

Z

+∞

χn (s|u|)∇H(su) · u ds Z +∞ χ′n (s|u|)|u|H(su) ds = χn (|u|)H(u) + 1   Z +∞ u ′ dt . χn (t)H t = χn (|u|)H(u) + |u| |u| = −

1

Hence, if |u| ≥ Rn + δn then Hn (u) = 0. If |u| < Rn + δn then Z     Z Rn +δn +∞ ′ u u ′ dt ≤ χn (t)H t χn (t)H t |u| |u| |u| Rn ≤ 2 sup |H(u)| . |u|>Rn

Therefore, since H(u) → 0 as |u| → +∞, the conclusion follows.

Step 2. λHn → λH . Thanks to the definitions given at the beginning of the proof, we have WH (u) = VJ (u) and WHn (u) = VJn (u) for every u ∈ X ∩ L∞ . Hence, for every u ∈ X with kuk∞ ≤ Rn one has WHn (u) = WH (u), and consequently λH,Rn = λHn ,Rn . Since λH,Rn → λH , in order to prove the thesis it is enough to show that λHn ,Rn − λHn → 0 .

(4.3)

To this aim, fix ǫ > 0 and let un ∈ X ∩ L∞ be such that D(un ) ≤ λHn + ǫ . |WHn (un )|

(4.4)

S2 -type parametric surfaces

13

Let π n : R3 → B Rn +δn be the retraction on the ball BRn +δn . By Lemma 2.4, since Jn (u) = 0 as |u| ≥ Rn + δn , WHn (un ) = WHn (π n ◦ un ). Moreover D(π n ◦ un ) ≤ D(un ). Hence also π n ◦ un satisfies (4.4). In other words, we may assume kun k∞ ≤ Rn + δn .

(4.5)

Now, set  v = τn u , with τn = min 1, n

n

Rn kun k∞



.

Note that kv n k∞ ≤ Rn and thus D(v n ) ≥ λHn ,Rn |WHn (v n )|. If τn < 1 then λHn + ǫ

1 D(v n ) τn2 |WHn (un )| |WHn (v n )| ≥ λHn ,Rn |WHn (un )|   ρn ≥ λHn ,Rn 1 − |WHn (un )| >

(4.6)

where ρn = |WHn (v n ) − WHn (un )| . Hence there exists θn ∈ [τn , 1] such that ρn

= =

(1 − τn ) |dVJn (θn un )un | Z 2 n n n n (1 − τn )θn Jn (θn u )u · ux ∧ uy R2

≤ (1 − τn )(Rn + δn )kJk∞ D(un )

≤ (1 − τn )(Rn + δn )kJk∞ (λHn + ǫ)|WHn (un )| .

(4.7)

By (4.5) and by the definition of τn , one has that (1 − τn )(Rn + δn ) ≤ δn and then, (4.6) and (4.7) imply λHn + ǫ > λHn ,Rn (1 − δn kJk∞ (λHn + ǫ)) . Hence, by the arbitrariness of ǫ > 0, and since λHn ,Rn ≥ λHn , one obtains 0 ≤ λHn ,Rn − λHn ≤ δn λ2Hn kJk∞ .

(4.8)

Now, we observe that WHn (u) → WH (u) for every u ∈ X ∩L∞ . This plainly implies that lim sup λHn ≤ λH . Then, since δn → 0, (4.8) implies (4.3). As a consequence of Lemma 4.2 we obtain the next result. Corollary 4.3 Let H ∈ C 1 (R3 ) satisfy (h3 ) and λH > 2. Then there exists a sequence (Hn ) ⊂ C 1 (R3 ) satisfying (4.1), (4.2) and (h1 ).

14

P. Caldiroli & R. Musina

Proof. Let H 0 (u) = H(u) − H∞ . By Lemma 4.2 there exists a sequence (Hn0 ) ⊂ Cc1 (R3 ) such that Hn0 → H 0 uniformly on R3 and λHn0 → λH0 . Take a sequence (Hn∞ ) ⊂ R \ {0} converging to H∞ , and set Hn (u) = Hn0 (u) + Hn∞ . Then (Hn ) satisfies the required properties, since λHn0 = λHn and λH 0 = λH . In the sequel we will also need the following property. Lemma 4.4 Let (Hn ) ⊂ C 1 (R3 ) be a sequence converging to some constant H0 ∈ R uniformly on B R . Then λHn ,R → +∞. R1 Proof. For every n ∈ N set mn (u) = 0 Hn (su)s2 ds and Qn (u) = mn (u)u. Then div Qn = Hn , and for u ∈ X, kuk∞ ≤ R one has |WHn (u)| ≤ k3mn − Hn kL∞ (BR ) R D(u) . Since for every u ∈ BR |3mn (u) − H0 | ≤ kHn − H0 kL∞ (BR ) , the thesis immediately follows. Now we study the behaviour of a sequence of minimal Hn -bubbles when the curvatures sequence (Hn ) approximates a given H. Theorem 4.5 Let H ∈ C 1 (R3 ) satisfy (h3 ) and let (Hn ) ⊂ C 1 (R3 ) be such that (i) Hn → H uniformly on R3 , (ii) λHn ≥ λ > 2 for every n ∈ N, (iii) for every n ∈ N there exists ω n ∈ BHn such that EHn (ω n ) ≤ µHn + ǫn , with ǫn → 0. 4π If kω n k∞ → +∞ then lim inf µHn ≥ 3H 2 . If lim inf µHn < ∞ ω ∈ BH with EH (ω) = µH . Moreover µH ≤ lim inf µHn .

4π 2 3H∞

then there exists

Proof. Suppose that kω n k∞ → +∞ and lim inf µHn < +∞ (otherwise the result trivially holds). By the assumption (ii) and by (3.1), one has (for a subsequence) sup k∇ω n k2 < +∞ .

(4.9)

˜ n (u) = Hn (u + pn ) and ω ˜ nLet pn = ω n (0) and set H ˜ n = ω n − pn . Then ω ˜ n is an H n bubble, with ω ˜ (0) = 0. Because of the conformal invariance, we may also suppose that |∇˜ ω n (0)| = k∇˜ ω n k∞ = 1. Moreover, thanks to Lemma 2.6, part (iii), the sequence of diam ω n is bounded, that is, there exists R > 0 such that k˜ ω n k∞ ≤ R. ˜ Furthermore, Hn → H∞ uniformly on B R and, by Lemma 4.4, λH˜ n ,R → +∞. Then, an application of Theorem 3.1 gives that ωn ) ≥ lim inf EH˜ n (˜

4π . 2 3H∞

S2 -type parametric surfaces

15

ω n ) = EHn (ω n ) ≤ µHn + ǫn , one gets But, since EH˜ n (˜ lim inf µHn ≥

4π . 2 3H∞

4π n Now, suppose lim inf µHn < 3H 2 and let (ω ) be the sequence given by (iii). Hence, ∞ up to a subsequence, one has

sup kω n k∞ < +∞ .

(4.10) n

n

Moreover, as before, (4.9) holds and one can assume |∇ω (0)| = k∇ω k∞ = 1. Therefore Theorem 3.1 can be applied again to infer that BH 6= ∅ and µH ≤ 4π lim inf µHn . In particular µH < 3H 2 . Finally we prove that µH is achieved. Note ∞ that (i) implies λH ≥ lim sup λHn (see the proof of Lemma 4.2). Hence, by (ii), λH > 2. Let (ω n ) ⊂ BH be such that EH (ω n ) → µH . We can apply the first part of this Theorem (with Hn = H for every n ∈ N), to deduce that (ω n ) satisfies (4.10). Moreover, since λH > 2, also (4.9) holds and a new application of Theorem 3.1 gives the conclusion. Proof of Theorem 4.1. It is a consequence of Lemma 4.2, Theorems 3.3 and 4.5. Now we discuss Theorem 1.3. We do not give the complete proof of Theorem 1.3 and we refer to [3] for all the details. Here we limit ourselves to sketch the procedure followed. Firstly one proves the result under the additional condition that H is constant far out. In this step, the main difficulty is to show that cH = µH , being cH defined by (1.2). To do this, one introduces a family of approximating compact problems given by  div((1 + |∇u|2 )α−1 ∇u) = 2H(u)ux ∧ uy in D (P )α u=0 on ∂D, where D is the unit open disc in R2 and α > 1 (close to 1). This kind of approximation goes in the spirit of a well known paper by Sacks and Uhlenbeck [8] and it turns out to be particularly helpful in order to get uniform estimates. More precisely, by variational methods, one finds that for every α > 1 (close to 1) problem (P )α admits a nontrivial solution uα ∈ H01,2α (D, R3 ). The family of solutions (uα ) turn out to satisfy the following uniform estimates: inf k∇uα k2 > 0, α

sup (kuα k∞ + k∇uα k2 ) < +∞, α

EH (uα ) → cH as α → 1. The limit procedure as α → 1 is a delicate step. Indeed the weak limit u of (uα ) is a solution of n ∆u = 2H(u)ux ∧ uy in D (P ) u=0 on ∂D.

16

P. Caldiroli & R. Musina

A nonexistence result by Wente [11] implies that u ≡ 0. Hence a lack of compactness occurs by a blow up phenomenon. One introduces the functions v α (z) = uα (zα + ǫα z) with zα ∈ R2 and ǫα > 0 choosen in order that k∇v α k∞ = |∇v α (0)| = 1. One can prove that there exists ω ∈ X such that v α → ω weakly in X and strongly in 1 Cloc (R2 , R3 ), and ω is a nonconstant solution of ∆ω = 2λH(ω)ωx ∧ ωy on R2 , for some λ ∈ (0, 1]. The assumption (h6 ) enters in an essential way in order to conclude that λ = 1 and EH (ω) ≤ cH . Again the condition (h6 ) gives that if BH 6= ∅ then µH ≥ cH . Hence one concludes that EH (ω) = µH . As a last step, one removes the additional assumption H constant far out following an argument similar to the proof of Theorem 4.1. We conclude this Section by showing a lower bound for µH under the condition (h6 ). 4π Proposition 4.6 Let H ∈ C 1 (R3 )∩L∞ satisfy (h6 ). If BH 6= ∅, then µH ≥ 3kHk 2 . ∞ The proof of Proposition 4.6 is based on the following argument. Since H ∈ L∞ (R3 ), by Lemma 2.1, the value SH = inf

u∈X

D(u)

2

|VH (u)| 3

(4.11)

is a well defined, positive number (apart from the trivial case H ≡ 0). By the definition (1.2) of cH , and by (4.11) one easily obtains that  3 SH cH ≥ . 3 As already mentioned, the condition (h6 ) implies that if BH 6= ∅ then µH ≥ cH . Hence the conclusion of Proposition 4.6 follows by the next estimate. Lemma 4.7 If H ∈ C 0 (R3 ) ∩ L∞ then √ 3 36π . SH = 2/3 kHk∞ Proof. As proved by Steffen in [9], for every u ∈ X there exists a measurable function iu : R3 → R satisfying the following properties:

(i) iu takes integer values; R 3 1 3 (ii) iu ∈ L3/2 (R3 ) and R3 |iu | 2 ≤ (36π)− 2 D(u) 2 ; R (iii) VH (u) = R3 iu H.

R Note that (i) and (ii) imply that iu ∈ L1 (R3 ) and then R3 iu H is well defined whenever H ∈ L∞ (R3 ). Using the properties (i)–(iii), we have |VH (u)| ≤ kHk∞ kiu kL1 ≤

S2 -type parametric surfaces

17

√ 3/2 −2/3 kHk∞ kiu kL3/2 ≤ (36π)−1/2 kHk∞ D(u)3/2 , and then SH ≥ 3 36πkHk∞ . Now let 3 δ,p us prove the opposite inequality. For every p ∈ R and δ > 0 let u = δω 0 + p, with ω 0 defined by (2.1). Note that D(uδ,p )

=

VH (uδ,p )

=

4πδ 2 Z

H(ξ) dξ .

Bδ (p)

Hence

√ 3 36π D(uδ,p ) = . δ,p 2/3 δ→0 |VH (u )| |H(p)|2/3 √ −2/3 By the arbitrariness of p ∈ R3 , one infers that SH ≤ 3 36πkHk∞ . SH ≤ lim

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