Equations of Mixed Type 0821845012, 9780821845011

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Equations of, Mixed Type

2.

Volume

Fite

: Translations of Math ета!

Translations

of Mathematical Monographs Volume 51

Equations of Mixed Type by

M. M. Smirnov

American Mathematical Society Providence, Rhode Island 1978

YPABHEHHH CMElllAHHoro TJ1TTA M.M.CMHPHOB J13tl,ATEJ1bCTBO "HayKa" MocKBa 1970

Translated from the Russian by Israel Program for Scientific Translations AMS (MOS) subject classifications (1970). Primary 35M0S; Secondary 76H0S. ABSTRACT. This book is devoted to the theory of partial differential equations of mixed type. The author introduces the reader to the current state of mathematical problems closely connected with transonic gas dynamics. Fundamental boundary value problems are considered: the Tricomi problem, the generalized Tricomi problem for Caplygin's equation, the Frankel' problem and the modified Tricomi problem.

Library of Congress Cataloging in Publication Data

Smirnov, Modest Mikha!lovich. Equations of mixed type. (Translations of mathem~ical monographs; v. 51) Translation of Uravneniia smeshannogo tipa. Bibliography: p. l. Differential equations, Partial. I. Title. II. Series. QA377.S5713 515'.353 78-8260 ISBN 0-8218-4501-2

Copyright © 1978 by the American Mathematical S~ciety

TABLE OF CONTENTS Chapter I. Introduction.......................................................................................... § 1. Brief review of results.............................................................................. §2. Canonical forms of second-order equations of mixed type...............

1 1 14

Chapter II. The Tricomi problem........................................................................ § 1. Statement of the Tricomi problem......................................................... §2. Extremum principle and uniqueness of the solution of the Tricomi problem............................................................................................. §3. Uniqueness theorems for the Tricomi problem for the Caplygin equation...................................................................................................... §4. Existence theorem for the solution of the Tricomi problem............ § 5. Further investigation of the Caplygin equation.................................... §6. The Neumann-Tricomi problem...............................................................

17 17 19 40

47 83 116

Chapter III. The generalized Tricomi problem................................................ § 1. Statement of the generalized Tricomi problem.................................... § 2. Uniqueness of the solutionsof the generalized Tricomi problem for for the Caplygin equation........................................................................ §3. Existence theorem for the generalized Tricomi problem..................

131 131

Chapter IV. The Frankel' problem..................................................................... § 1. Statement of the problem........................................................................ § 2. Uniqueness of the solution of the Frankel' problem........................ §3. Existence of the solution of the Frankel' problem...........................

183 183 184 188

132 140

Chapter V. The modified Tricomi problem for an equation of mixed type............................................................................................................. § 1. Statement of the problem........................................................................ § 2. Uniqueness theorem................................................................................. § 3. Some results from the theory of derivatives and integrals of fractional order for a real function of one variable........................... §4. Investigation of equation (1.1) in the hyperbolic half-plane............ §5. Investigation of equation {1.1) it1 the elliptic half-plane.................. §6. Existence theorem for the Tricomi problem.......................................

200 201 207 210

Bibliography...............................................................................................................

222

iii

197 197 198

CHAPTER

I

INTRODUCTION § 1. Brief review of results

A partial differential equation is said to be an equation of mixed type if it is elliptic in one part of the domain and hyperbolic in the other, both parts being separated by a transition curve ( or surface) upon which the equation either degenerates into a parabolic equation or is undefined. The theory of equations of mixed type has a comparatively short history. The first really deep results were published by Tricomi in the 1920"s. He studied the fundamental boundary-value problem (Tricomi problem) for the equation {1.1) now called the Tricomi equation. Let the domain D be bounded by a smooth curve r with endpoints A and B on the x-axis, and two characteristics 11 and 12 of equation (1.1), emanating from A and Band intersecting at a point C of the lower half-plane. The problem is to find a solution, regular(') in D, satisfying the boundary conditions u = I{) on r and u = t/J on 11 . Tricomi [69a) proved the existence and uniqueness of the solution, on the assumption that the curve r ends in two arbitrarily small arcs AA' and BB' of the normal curve (x - x 0 )2 + 4y 3 /9 = c2 , lying outside the latter curve elsewhere. Under these assumptions, Tricomi reduced the problem to that of finding a function v(x) = ou(x, 0)/oy satisfying the singular equation

(l)A function u(x, y) is a regular solution of equation (1.1) if: 1) it is continuous in 15; 2) its first derivatives are continuous in i5, with the possible exception of the points A and B, where they may have poles of order less than 1; 3) its second partial derivatives are continuous in D, with the possible exception of points on the parabolic curve, where they may not exist; 4) it satisfies equation (1.1) at all points of D\ AB.

1

I. INTRODUCTION

2

f( I

v(x)

I I + nVa" xt )21, ( t-x 0

I

- t+x-2xt

) 'V (t) dt

(1. 2)

I

+

f

.K (x, t) v (t) dt

= F (x),

0

where the kernel K(x, t) is of Fredholm type. In his doctoral thesis (24a] , Gellerstedt solved the Tricomi problem for the equation

a2u a2u -cu=F(x, y ) y..m_+ax2 ay2

(1.3)

under the same assumptions concerning r as adopted by Tricomi. He then (24c] studied boundary-value problems for the equation

(1.4)

A

FIGURE 1.

(m is an odd natural number), where the boundary values in the hyperbolic part of the domain D are prescribed either on two characteristic arcs PC1 and PC2

which issue from an interior point P of the interval AB, or on characteristic arcs AC1 and BC2 (Figure 1). In the elliptic half-plane, the boundary values are prescribed on the curve

x2 +

(m

+4 2) 2

y O).

Following Tricomi, Gellerstedt reduced these problems to a singular integral equation and subsequently, using Carleman's method [ 11] , to a Fred.hold equation. A new stage in the theory of equations of mixed type was initiated by Frankl' [22a]. He showed that the problem of a supersonic jet flow out of a

§1. BRIEF REVIEW OF RESULTS

3

plane-walled vessel (the velocity inside the vessel is subsonic) can be reduced to the Tricomi problem for the Caplygin equation 2u a2u K (y) a ax2 + ay2 = 0

(K (0) = 0,

K' (y) > 0).

(I .5)

In another paper [221], Frankl' elaborated an important generalization of the Tricomi problem, in which the boundary values are prescribed on the curve r and on a noncharacteristic arc AE lying inside the characteristic triangle and intersecting each characteristic of the second family at most once. The point E lies on a characteristic BC of equation (1.5). Henceforth we shall call this the generalized Tricomi problem. Vekua [72] has pointed out the importance of equations of mixed type for the theory of infinitesimal deformations of surfaces and the zero-moment free theory of shells with curvature of variable sign. The equations of magnetohydrodynamic flows in the region of sonic and Alfven velocities [42], and the equations of the flow of water in an open channel at a velocity exceeding the propagation velocity of surface waves [22d], are also of mixed type. In order to simplify the investigation of boundary-value problems for equations of mixed type, Lavrent'ev [47] suggested considering the model equation

a u +sgny aay2u 2

ax2

2

=0.

(I 6)

.

A detailed investigation of the Tricomi problem and its generalizations for equation ( 1.6), under quite general assumptions concerning the curve bounding D in the upper half-plane, was carried out by Bicadze [I Oa-d, i]. He also solved the Tricomi problem for equation (1.6) in a multiply-connected domain with au/an prescribed on I'. Two applications of boundary-value problems for equation (I .6) to gas dynamics are presented in Frankl' [22b] . Using the properties of simple automorphic functions, Cibrikova [I 2b] obtained a solution of the Tricomi problem for equation (I .6) in explicit form, doing this without use of conformal mappings when I' is half the boundary of one of the fundamental domains of an elementary or Fuchsian group of Mobius transformations. The following extremum principle holds for the Tricomi problem. A solution of the Tricomi problem vanishing on the characteristic AC achieves neither a positive maximum nor a negative minimum on an open arc AB of the typedegeneracy curve. This principle was first stated by Bicadze [10a] for equation (1.6). Somewhat later, the principle was established for the Tricomi equation (1.1) by Germain and Bader [26b]. Babenko [4a] proved the principle for the Tricomi problem when the equation is

I. INTRODUCTION

4

iJ 2 u

Y ax2

au au au + ay2 +a(x. y) ax +b(x. Y)ay+c(x. y)u=O 2

assuming that a(x, 0)

= b(x, 0) = 0

(1.7)

and that the transition curve is sufficiently

small. The extremum principle is of great importance, first, because it immediately implies uniqueness of the solution, and, second, because it enables one to apply the alternating method of Schwarz to solution of the Tricomi problem under quite general assumptions on r. Also worthy of mention here is an interesting maximum principle (Agmon, Nirenberg and Protter [3]) for the equation

K (y) Uxx + Uuu + a (x, y) Ux + b (x, y) Uy+ c(x. y) u = 0,

(1.S)

where K(y) > 0 for y > 0, K(y) < 0 for y < 0, and K(0) = 0. Suppose that a twice continuously differentiable solution u(x, y) of equation (1.8), nondecreasing as a function of y along one of the characteristics, is defined in the characteristic triangle ABC. Suppose moreover that the following conditions hold in the characteristic triangle:

6 (V - K)

+ a + b V- K ~ 0,

1 [6(V- K) +a+ b \f - Kl 6( 6 (-v=R) + a+ bV=-R)+-

V-K

Jf--=-R

2K

X[6(\f-K)+a-b~]-2c~o.

c~O,

where

Then, if the maximum of u(x, y) is positive, it is attained on the interval AB of the parabolic curve. The uniqueness theorem for the Tricomi problem for equation (1.8) follows easily from this maximum principle. Pul'kin (64c] proved that the generalized solutions{2) of the Caplygin equation satisfy this maximum principle. A further generalization of this result for equation (1.7) is due to Volkodavov (73a]. The first proof of a uniqueness theorem for solutions of the Tricomi problem for the Caplygin equation (1 .5) is due to Frankl' [22a]. He assumed that the function

F (y)

= 2 ( :, )' + I

( 2)For a definition of generalized solutions, see p. 33.

(I .9)

§ 1.

BRIEF REVIEW OF RESULTS

5

is nonnegative for y < 0. Note that this condition holds everywhere in the case of the Tricomi equation (1.1). Protter [63b J generalized the uniqueness theorem, allowing F(y) to be negative for y < 0. He also proved [63c] a uniqueness theorem on the assumption that K(y} has a continuous third derivative such that K"' < 0 for y < 0. In his uniqueness proof, Protter used the so-called a, b, c-method, based on an idea of Friedrichs. The essence of the a, b, c-method is as follows. Let u(x, y) be a quasi-regular solution(3) of equation (1.5) defined in a domain D, and satisfying homogeneous boundary-value conditions. Consider the integral 0

f f (au+ bu.x + cuy) (Ku.xx+ Uyy) dx dy, D

where a, b and c are sufficiently smooth functions of (x, y). By virtue of (1.5), this integral vanishes. The functions a, b and c are chosen in such a way that, after transformation of the integral by Green's formula, one obtains a positive definite expression which vanishes only if u(x, y) = 0. The a, b, c-method was also used by Wu and Ding (79] and Wang (76] in order to prove a uniqueness theorem for the Tricomi problem for the Caplygin equation. Babenko (4a] carried out a detailed investigation of the Tricomi problem for the Caplygin equation (I .S). He proved the existence and uniqueness of a generalized solution of the Tricomi problem for this equation, on the assumption that ldx/ds I < C), 2 (s) in a neighborhood of the points A and B on r, where C is a constant and x = x(s), y = y(s) are parametric equations of r. For the Tricomi equation (1.1), Babenko was able to drop the restrictions on the behavior of r near the axis y = 0. Pul'kin [64b] investigated the Tricomi problem for the equation

a2u

a2u

ax2 + sgn Y ay2

au au + a (x, y) ax + b (x, y) ay + c (x, y) u = 0.

(1 IO)

·

He proved the existence and uniqueness of a generalized solution to the Tricomi problem, assuming that the tangent to the smooth curve r at A and B is parallel to the y-axis, and imposing certain restrictions on the coefficients a, b, c. Vostrova [75b] proved the existence and uniqueness of the solution of the boundary-value problem for equation (1.10), the values of au/an being prescribed on r and those of the unknown function on a segment of the characteristic x y = 0. Krikunov [44] considered the problem for equation ( l.l 0) with derivatives in the boundary condition on r, assuming the coefficients analytic in the upper half-plane. Smirnov [66e] investigated the boundary-value problem for (3)For the definition of quasi-regular solutions, see p. 41.

I. INTRODUCTION

6

equation ( 1.4), with the quantity m

dy au

dx au

Y Ts ax - Ts Ty prescribed on

r

and the values of the unknown function on the characteristic

1•.

Karatoprakliev [39) grneralized the Tricomi problem for equation (I .1) to the case that the solution u(x, y) and its derivative au/an may have a discontinuity of the first kind on the parabolic curve and satisfy there the following "matching" conditions: u(x, -O)=a.(x)u(x, +O)+v(x), au(~ -0) y

= ~ (x)

au(~ +o) y

+ 6 (x),

-l 0 equations (2.2) are elliptic; for y < O they are hyperbolic. Consider the simplest and, at the same time, very important case of power degeneracy: in a neighborhood a of 'Y, the function d(x, y) may be expressed as ll (x, y) =

Hn (x, y) M

(x, y),

(2.3)

§2. CANONICAL FORMS OF EQUATIONS

15

where M(x, y) is a function nonvanishing and finite-valued in a; H(x, y) = 0 is the equation of r, Hx and HY do not vanish simultaneously, and n is an odd positive integer.( 4 ) Since by assumption the function ~(x, y) changes sign across r, it follows that for all odd n equation (2.1) is elliptic on one side of r and hyperbolic on the other side, i.e. it is of mixed type. Tricomi [69a] and CinquiniCibrario (13a) showed that. under certain restrictions on the smoothness of A, Band C, equation (2.1) may be reduced in a neighborhood of r by a nonsingular real transformation of the independent variables (we retain the previous notation for the new independent variables) to the canonical form

(2.4) if (2.1) is an equation of mixed type of the first kind, and to the form

(2.5) if it is of the second kind. Of special interest are the linear equations

+ "1111 + al (x, y) Ux + bi (x, y) u11 + ci (x, y) u = f i (x, y), Uxx + y"'u/lY + a2 (X, y) Ux + b2(X, y) Uy+ C2(X, y) U = f2(X, y).

YnUxx

(2.6)

In addition to the above examples, equations (2.1) with a power degeneracy of arbitrary order a > 0 are of great interest. For these equations we have A(x, y)=sgnH(x, y)IH(x, y)la M(x, y). The simplest examples are

sgn yly Uxx

In particular, if a

r

Uxx

+ Uy11 = o,

(2.7)

+ sgn yly (' u1111 = 0.

= 0 this gives the even simpler equation Uxx

+ sgn YU11g = 0.

(2.8)

We make special mention of the Caplygin equation, which is of paramount importance in gas dynamics of transonic flows. This equation for the stream function has the form

K (o) '¢ea+ il'aa = 0, where K(a) is a monotone increasing function of a and K(O)

=

(2.9)

= 0.

(4)When ,, = 2m (m 1, 2, 3, ..• ) we get degenerate equations of elliptic-parabolic or hyperbolic-parabolic type.

I. INTRODUCTION

16

The substitution

reduces equation (2.9) to canonical form:

Y'i'xx + 'i'1111 + b (y) 'i'11 = 0,

(2.10)

where

b (y) = Setting 1/1

= u exp(-¼ fl

d 2 y • ( dy dcr 2 • Ta

b(y) dy ), we bring (2.10) to the form

YU xx+ Uyy

where c(y)

= -¼[b 2 (y) +

)2 '

2b'(y)].

+ C (y) U = 0,

(2.11)

CHAPTER

II

THE TRICOMI PROBLEM § 1. Statement of the Tricomi problem

Consider the equation a2 u a2u au au K(y) ax2 ay2 +a(x, Y)ax+b(x, Y)7iy

+

+ c (x,

y) u = f (x, y),

(1.1)

where K(y) is a continuous monotone increasing function of y and K(0) = 0. Equation (1.1) is elliptic for y > 0 and hyperbolic for y < O; y = O is its parabolic curve.

C FIGURE 3

Let r be a rectifiable Jordan curve lying in the upper half-plane with endpoints at A (a, 0) and B (b, 0). For y ·< 0, (1.1) has two families of real characteristics, defined by the equations dy = - - -I - ,

dx dy

(1.2a).

V-K(y)

I

-;i; =Ji-;=_=K=(=y=-)

(1.2b)

Let D denote the domain bounded by r and by characteristics AC and BC belonging to families (1.2a) and (1.2b), respectively (Figure 3). The region of Din the half-plane y > 0 (y < 0) is denoted by n+ (DJ. 17

II. THE TRICOMI PROBLEM

18 TRICOMI PROBLEM.

Find a solution in D of equation (1.1), continuous in

D, which assumes on the curve rand on one of the characteristics, say AC, prescribed continuous values ulr = ip(s) and ulAc = 1/J(x), with ip(/) = 1/1(0), where l is the length of r. On the parabolic curve y = 0 of equation (1.1), we have matching conditions

lim u (x, y) = lim u (x, y)

y ➔ +O

. au ,. au Ilffi - = lffi ay y ➔ -o ay

y ➔ +u

FIGURE 4

(a~ x ~ b),

y ➔ -U

(a (y < 0) which satisfies equation (2.1) in the half-plane y < 0 and the initial data

ul ,-o = 't (x),

!" I

y 11-0 = 'V

.

(2.2)

In characteristic coordinates

s=X- 32 (-y)'/', equation (2.1) takes the form

where

and

The initial data (2.2) become (2.4)

It. THE TRICOMI PROBLEM

20

A solution of (2.3) satisfying conditions (2.4) is given (see [4a]) by 'I')

'I')

J g (s,

u (s, 11) =

J h (s, 11;

11; t) -r: (t) dt -

s Here

s

½(4r [2 (/f/

h (s, 11; t) =

c2 .s)

t) v (t) dt.

\ 6

(t - s)- 11' (11 - t)-'is

{

t

+ f u - s')-' 1, (11 -

t)-'1s

B1 (s',

11)

s1 (s';

s, 11) ds'

s

t

+f

(2.6)

(11'-t)-'1'(t-'s)- 11'A1(s,

11')Ai(r]'; s, TJ)d11'

'l')

+ff t

g('s

(11' - t)-'l,(t - s')-' 1'D [v (s', TJ';

s, 11)] ds' dr}'),

'l')

· t)-

'11,

- -

r(l/a){

b(t O)h(t · t)+ ' '' (T) ~ s>'i, c (s', ,,,: s,

(T)' _

11),

11) is a bounded function.

Frankl' [22k] proved Lemma I for O < 11 -

~

< e, where e > 0 is

(2.12)

II. THE TRICOMI PROBLEM

22

sufficiently small. Babenko [4a] showed that the lemma is valid without any restrictions on f/ - ~. It can be shown [22k] that if r(t) and v(t) are twice continuously differentiable for x 1 < t < x 2 , then the function u(~, T/) defined by (2 .5) has continuous derivatives up to second order with respect to x and y in the domain bounded by tlie characteristics ~ = x 1 , y = x 2 and the parabolic curve f/ = t. In this case we have the classical solution of the Cauchy problem (2.1)-{2.2). The expression (2.5) will be called a generalized solution of equation (2.3) if r(t) and v(t) are continuous for x 1 < t < x2 . A necessary condition for a generalized solution to possess some smoothness property is that the functions r(t) and v(t) be smooth in a suitable sense. Henceforth, we consider the generalized solution in the triangle ABC bounded by the segment AC of the characteristic ~ = 0, the segment CB of the characteristic T/ =land an arc of the parabolic curve T/ = t (!:;ABC= {t T/ E (0, /], T/ > U). Consider the following class of generalized solutions of the Cauchy problem introduced by Babenko [4a]. CLASS R 1. A generalized solution (2.5) of equation (2.3) belongs to class R 1 if T(t) satisfies a Holder condition with exponent a 1 > 5/6 for 0 ,s;; t 1/6 for 0 ,s;; t < l. Below we shall often use expressions of the type X

r:a) J(x-t)

fa(x)=

0- 1

f(t)dt

a

where f(t) is integrable on (a, b ); such integrals are known as integrals of fractional order a. (4a]. If a generalized solution u belongs to the class R 1 , then its derivatives ux and uy are continuous in MBC, uy is continuous right up to the parabolic curve, and LEMMA 2

Iim~= v(x)

y➔ O

PROOF.

oy

(0 0 and u(Q) > u(P), then (~1U11)Q < (~1U 11 )p.

PROOF. According to our assumption, L1 (u) ,;;;; 0 on rP, Ql; hence on ( 3 ) (131 ufl)Q denotes the value of /Ji ufl at the· point Q.

[P, Q].

§2. EXTREMUM PRINCIPAL AND UNIQUENESS

37

Integration along [P, Q] gives Q

J l(P1uTJ)s + (a1u), + y1u] d6 ¾ 0, p

and thus

Q

(P1uTJ)Q - (P1uTJ)P ¾ -

f y 1u d; + (a u)p 1

(a 1u)Q.

(2.54)

p

In the right-hand side of (2.54) we add and subtract Q

u (Q)

Q

f V1d; = u (Q) JP1V d6 -

p

u (Q) [a 1 (Q)- a 1 (P)J,

p

so that we have Q

f [u (Q) - u] y 1d6 u (Q) f a1v d; - [u (Q)- u (P)I a1(P).

(P1uTJ)Q - (P1uTJ)P ¾

p

Q

-

(2.55)

p

Using conditions (I) and the fact that u(Q);;;;,. u(P), we have

(J31u,)Q ~(J31urr)P. Therefore, if u(Q) > 0 and u(Q) > u(P), then at least one term on the right of (2.55) is negative, by conditions (I'). Consequently,

(P1ll11>Q < (P11tTJ)p-

Q.E.D. The proof of Theorem 2.3' follows easily from Lemma 6. Indeed, suppose that the maximum of u(t 11) in D1 is not attained on AB. It cannot be attained on AC(~

= 0) either, since max u on

AC is attained at the

point A (as urrC0, 11) ~ 0), which belongs to AB. Thus the maximum point Q lies in D + CB. Let the straight line 1/ = const through the point Q cut AC at a point P. The maximum of u on [P, QJ is attained at the point Q; and u{Q) and u(Q)

> u(P).

(P111'1>Q

and since ur,CP)

~

>0

By Lemma 6,

0, we get ur,CQ)

< (P1uTJlp,

< 0.

But then Q cannot be a maximum point

of u in D1 . Q.ED. Now, using Theorem 2.3 1 , we shall prove Theorem 2.3.

Let g(t 77) be a function satisfying the conditions

Mi(g)==g, 11 + ag~ + pgTJ > 0 in

D1,

(2.56)

II. THE TRICOMI PROBLEM

38

(2.57) Such a function is easily constructed (for example, g(t 1/) = sinh k~ cosh k1) for sufficiently large k). If u is a function satisfying the conditions of Theorem 2.3, consider the set of functions ue = e-Eg u, e > 0. It easily is seen that

L 1e (Ue ) =

i} 2 ue ' OUe 05 OT] T 0, and by the characteristic arcs y

v,:

x= -

y

f V- K(t)dt,

v2: x - a=

0

f V- K (t) dt

(3.2)

0

of equation (3.1). The Tricomi problem consists in finding a solution of (3.1) in the domain D which has prescribed values on r and on the characteristic 'Y 1 • Frankl' [22a] proved a uniqueness theorem for the Tricomi problem for equation (3.1), assuming that the function

F (y) = 2 (

i' )' + 1

(3.3)

is nonnegative for y < 0. Protter [63b] generalized this theorem, including cases in which F(y) may be negative for y < 0. We shall prove uniqueness theorems for the Tricomi problem for two types of domains D. In the frrst case the curve r lying in the elliptic half-plane is arbitrary, but the part of the domain located in the hyperbolic half-plane is restricted by the condition F(v) > d, where d < 0. In the second case the hyperbolic part of the domain is arbitrary and F(y) may have arbitrary values for y < 0, but r cannot extend too far in the direction of the y-axis. DEFINITION. A function u(x, y) is said to be a quasi-regular solution of equation (3.1) in the domain D if it belongs to class c< 2 >(D) n C(D), satisfies (3.1) in D, the integrals a

Ju(x, O)uy(x, O)dx, f J(Ku~+u~)dxdy

o

n+

exist, and Green's formula is applicable to the integrals

f f ul (u) dxdy, f f D

D

UxL

(u) dx dy,

f f uyl (u) dx dy. D

The contour integral in Green's formula is the limit of integrals along curves approaching the boundary of the domain D from its interior. THEOREM 2.6 [63b]. Assume that: I) K(y) is a monotone increasing function with continuous second derivative, such that K(0) = 0, K'(y) i= 0 for y < 0, .F{0) > 0; 2) there exists a constant d 1 < 0 such that F(y) ;;;. d 1 in D; and

42

II. THE TRICOMI PROBLEM

3) u(x, y) is

ii

Then u(x, y)

= 0 in D.

THEOREM

quasi-regular solution of equation (3 .1 ), vanishing

011

r

and

r1.

2.7 [63b]. Assume that: 1) K(y) satisfies condition 1) of Theo-

rem 2.6: 2) there exists a constant d ~ > 0 such that y,,, < d 2 , where y,,, is the maximum of the ordinares of the points of r. and 3) u(x, y) is a quasi-regular solution of equation (3.1), vanishing on rand r 1 • Then u(x, y) = 0 on D. Theorems 2.6 and 2.7 will be proved simultaneously. Consider the integral 'Y1

JJ(au+ bux + cuy) (Kuxx + llyy) dx dy,

(3.4)

D

where a(x, y). b(x, y) :mJ c(x, y) :ire suffi.:iently smL1L1th fun.:tions and u (x, y) is a quasi-regular solution of (3.1).

Applying Green's formula to the integral (3.4), we get

0= •1·1·[_!__(Ka +a !IY )u 2 -a(Ku2X +uY2 )-_!__b (Ku 2X -u2y ) • 2 .u 2 X D

-bu u yxy

+

r

•. r+-r 1+-r2

_ _!__(cK) u2 -c x Ku xuy _ _!_c u2 ]dxd11• 2 yx 2 yy

r-

arm y

+

1 c(Ku! - u2 21 a y u2 - buxu y +-2 g

l~t the solution u(x. y) \·a.nish on

r

and

r 1.

Take b

)l dx

= c = O in D+.

12 can be written as 12

=J· (- bu u + X

y

1 2

c(Ku 2X -u2y,)]dx

+[~ b(Ku!-u!)+cKuxuuldy + J[- a11uy + ½ayu 2 -

buxuy

+ ½c(Ku!- u!)] dx

'Y2

+ rlaKuu,, or, using the equalities

I I J 2 axKu 2 + 2 b (Ku!- tt!) + cK11x11 Y dy

Then

43

§3. UNIQUENESS THEOREMS FOR THE CAPLYGIN EQUATION

as

12=

½f (b-c V- K)(\/- Ku.x-uy)(u.xdx+uvd!J) 71

2 +2uu +__l_-u2 )dx -ff(b+cf-K)(V-Ku ~ x xg J-1-K g

-

f

(3.5)

'Y2

a

l r - -Ku du - 2I l r-- K u2 (axdx + aydy) = 11 + 12 + la1

-

72

Since u = 0 on i' 1 • it follows that u xdx + uydy = 0 on i' 1 , and hence / 1 = 0. The integral J I can be divided into three integrals in the following way: / 1

Jf a(Ku;+u!)dxdy -{ f J{I2aK + Kbx - (cK)

= -

D+

D-

+ (2a - bx+ c11 ) u!} dx dy

11 J

u; + 2 (Kcx + by)uxu 11

+ff (Kaxx + a

11 y)

(3.6)

u 2 dx dy

D

=l4+Is+I6. We claim that it is possible to choose the functions a, b and c so that all the integrals / 2 , . . . , / 6 are non positive. As their sum is equal to zero, it will follow that each of them must vanish. Then / 4 = 0 (for a > 0) will imply that u(x, y) = c inn+, and since u = 0 on r, it will follow that u(x, y) = 0 inn+. In particular, u(x, 0) = 0 and au(x, 0)/ay = 0, so that u(x, y) = 0 inn-, because of the uniqueness of the solution of the Cauchy problem for equation (3.1). Thus, u(x, y) = 0 everywhere in D. For y < 0 we set

4aK

(3.7)

c=y,

Obviously, with this choice of b(x, y) we have / 2 = 0. Integration of / 3 by parts gives

!3 =

f (v-- K ax+ ay +

aK') 2 K u dx. 4

This integral is nonpositive if for

y' •

s')

aTJ

I

ri ·

Hence, in view of the inequalities

Iav

s, 11)

(s', TJ'; a11

I< I I

C

v 5

TJ -

s

for

6

11-TJ'

for

C

v

s, ri' < s, ri' >

(4.52)

simple manipulations give

I

~ aL(s, TJ, t) aTJ

l t).

J/3

0

(4.64)

Next, the substitution z 3 = (x - ~)/(t - ~) gives

x>t, (4.65)

x() '-V

X

k(2)'/, (-)

=2 3

(j)

~

X

-'l(l 3

X

-

)

f2 (~)

3 F(2

f ( {)

7 4.

2x-l)

3' 6' 3 ' -x-2

'

Hence. using the relation between hypergeometric functions with arguments z and 1/(1 - z) (see [49], formula (9.5.8)), we have

I.

lfJ:""lxl°'lf(x)IP dx


0,

and let 0 1 ( E) and 0 2 (E) be the abscissas of the intersection points of 'Y and Tl = 1 with the line a = -c let BE be the intersection point of Tl = l with the line a = -E.

Integrating (5.63) over the domain e, (e> 2 pu 0 (0, - e) u0 (0, - e) d0

J

v-:;

0,

we get

+ JP [ -

~w

2u8u0 d0

+ (Kui- u;) dcr]

~

Be

f P ( l) (-V- K ua + u )2 dcr + Jf p' (11) ( V- K u + u d0 da = 0. 0

CV

8

0)

(5.64)

2

DYE By (5.59), lim 0 ___ 0 q 0 ~ O; thus, it follows from (5.58) that p'(B, 0) ~

= p'(B)

0. Set

p(l) = By Lemma 5, we can let

o.

(5.65)

0 in (5.64), to obtain

€ -

I

2

f puauad0+ f p[-2uauad0+(Ku~-u;)dcr] +ff p'(ll)(V-Ku +uo)2d0dcr=0.

Q

'V

0

v; Subtracting (5.62) from (5.66), we get

(5.66)

II. THE TRICOMI PROBLEM

108

f p[-2u uad0+(Ku:-u!)da] =-ff p'(TJ)(Y-Ku +u .)2d8de1 8

V

8

0

D-:;

+ ff

(5.67)

f (Kuij + u!) (p da + q d0).

qKauida da -

r

D+

This equality will be of great in).portance in what follows. Let "f coincide with the characteristic ~ = 0. Along this characteristic,

d0 + V - K da = 0,

d'l'J = - 2 v=-7( da, (du)2

- 2u8ua d0 + (Ku~ - u~) da = - ~ = 2u~

v- K dri.

Then it follows from (5.67) that I

2

J p (11) u~ V - K dri = - JJ p' (ri) ("V="R u

0

+ ua)2 d0 da

D-

0

+ JJqKau~d0 da D+

f (Ku:+ u~) (p da + q d0).

(5.68)

r

Hence I

2

f p (TJ) u~ V- K dri~ JJqKau~d0da.

(5.69)

D+

0

There exists an infinite set of pairs of functions p and q satisfying (5.58), (5.59) and (5.65). Correspondingly, there is a class of curves r on which (5.61) holds. In particular, consider the pair of functions p = 1 - 0, q = a. Condition (5.61) becomes (5.70) where sis arc-length on r, 0(0) = 1 and a(0) = 0. In particular, any curve r whose union with its reflection in the a-axis is a convex curve satisfies (5.70). Throughout the remainder of this section, we consider only curves r satisfying (5.70); this will not be mentioned again. Returning to the solutions un, we have I

f 0

(l - ri)

Y-

K ( 0;;

YdTJ~; ff aKa( 0 )2 d0da. 0;

D+

§5. FURTHER STUDY OF THE CAPLYGIN EQUATION

109

In view of the relation between un and zn, we get y

-aun -=e a1} Since

R

< Cr/ 13

_ _I_ 2

J

b fy)dy

(

o

az

_ n - -b-(y) ·-'Yl-'l,z )

al]

2. 5'/s

< C1 , it

and exp(-1 /2 Jtb(v) dy)

n •

'I

follows that

(5.71)

By (5.37), we infer from (5.47) that

.r

(I - l]) TJ 11 ' (

0

~~n rd11 ~ d [i (I -

j

riH-Tn (ri) dl] +

0

(I - 11)

µ;, (11) d11]

O

I

~d

f

(l - ri)µ~(ri)dri.

0

Thus I

f

I

(1 - 11) TJ 11 ' (

~~

-

:~

)2 dl] ~ d

0

f

(I - l]) [µn (TJ) - µ (11)12 d11.

0

By Hardy's inequality (see p. 84),

By virtue of the last two inequalities, we can let n -

00

in (5.71). We

have proved LEMMA

6. If z is a solution of equation (5.5) in class j, vanishing on

then I

J(1 -

TJ) TJ'I, [

oz:~!]) - : _ 0 and 7(;)-7(t) > 0, we get v(s) > 0. Titls contradicts Lenuna 5 of §2. The extremum principle immediately implies the uniqueness of the solution of problem TN. 2. Problem Find a regular solution u(x, y) of equation (6.1) in D+, continuous in 75+, having continuous derivatives u and u in D+ U r which .I: I/ may have poles of order less than 2/(m + 2) at the points A and B, and satisfying the boundary conditions

ri.

A [u) s

= ym dyds ~ iJx

..!!!.._ iJu ds iJy

Ir = IJl (s)

(6.12)

§6. THE NEUMANN-TRICOMI PROBLEM

119

(6.13) The curve r is assumed to satisfy two conditions: I) condition I of §4; 2) the inequality ldx/dsl ¾ C 2 ym+t(s), where C is a constant, holds in a neighborhood of the points A and B. The uniqueness of the solution of problem

Ti follows easily from the for-

mula

ff (ymu! + u!) dx dy n+

J uAs [u] ds,

=-

(6.14)

r+AB

where u is a solution of ( 6 .1 ). In the elliptic half-plane, (6.1) l\as a fundamental solution Q2 (x, y; Xo, Yo)

s:k2(

m~ 2

t- (rfr 2

13(1-0) 1- 2PF(l -~. 1-~. 2-2~; 1-a),

(6.15)

where

,22 }... (X -

Xo

)!

r1 ,2

(s) G2 (s, 'I'); Xo, Yo) ds,

0

where -r(x) and ',O(s) are continuous on [O, I], is a solution of problem T"J, in the domain D+. PROOF. The first integral in (6.50), call it / 1 (x 0 , y 0 ), is a solution of (6.1) continuous in [j+. By (6.28) and (6.39), it can be written as I

f 1 (xo, Yo)=

f +ff T

(x) aq2 (x, ~~ Xo, Yo) dx (6.51)

()

I

/

T

(x) A (t; Xo, y0) aA (~ x, O) dx dt. y

0 0

We claim that

(O < Xo < 1).

lim / 1 (xo, Yo)=,; (Xo) y, ➔ O

(6.52)

It is easy to see that the second integral in (6.51) vanishes for y 0 = 0, 0 < x 0 Thus we must prove that the first integral, i 1 (x 0 , y 0 ) say, tends to -r(x 0 )

< I.

asy 0 -+-0 (0

O and (3 > 0 such that a< yK/y < f3 everywhere in D. ~ = I -\.1--=-7!!_!!_ dx Y dx and so 1

'

< dr,/dx..;; 2; similarly, ~ dy

so that ldr,/dyl > 2..;Cy. We assume that

- -dx~ C dy"""

Then

Next,

f-

dx dy

v-y

'

l

y'

< C < oo.

(3.7)

III. THE GENERALIZED TRICOMI PROBLEM

142

2

V-

y

(0)

= 0, it

"'i,(J)'' (")

dar + (j (I- ")"-'I, (110)\ Tio

147

+-I JTJ Tio

0

l(11)I d

0

TJ'

}. l]

I

As before, we find that I

f

I

(1 - ri)

l1 11' (

0

!~ )2 d11 ¾ Ai f (1 - 11)1-i11' 0, it follows

Zn

(x) -

(x, 0)

~

=

't'n

(x),

b (0) Tn {x)] ::

lo -o •

that

I

f Tn(x)[vn(x)-

b~O)

Tn(x)]dx..11;:;;. m'IIM. Letting

n _.

1

00 ,

1

we get

II B*"' II~ m'II"' 11. Thus the equation B*>..

=0

has no nontrivial solutions in

fiO,

!). But then,

by a standard theorem of functional analysis, equation (3 .97) is solvable for any

/Ef iO, 1). We have thus proved LEMMA

4 [4a] . Let

r satisfy the condition dcr

( 1 - 0} ds

d0 + a Ts~ 0,

(3.98)

§ 3.

EXISTENCE THEOREM

181

and let conditions (3.2) hold along -y. Then equation (3.12), i.e. YZxx

:R in D,

has a solution z E zl'Y = (17), if

+ Zxy + C (y)z =0,

satisfying the boundary conditions zlr = 0 and I

I (l - 11) 11'

1• 0. Without loss of generality, we may assume that ip(O) = 0. The function (see (4.16) of Chapter II) l

z 1 (x, y)=-

Jq,(s)p(s, x, y)ds

(3.100)

0

is a solution of (3.12), continuous in l5+, satisfying the boundary conditions

. m az1 I1 -= 0 ay

(O 0, and D- the region of D in which y < 0. Inn+, the function F(_x, y) satisfies

K (y) F xx+ F uu = K' (y) u2• Using (2.4) to express u 2 in terms of the derivatives of F(x. y), we get

K (y) Fxx+ Fuu = aF x + ~F y,

{2.5)

where K' (y) Fx a=--;==== 2 KF2X +F2y '

V

The coefficients a: and ~ are bounded in any closed subdomain of n+, and equation (2.5) is elliptic inn+. Consequently, F(x, y) satisfies the maximum principle in the domain n+, i.e. it achieves its maximum on the boundary of n+, and at the point where this maximum is achieved {2.6)

~ maxD x, and for y ¾ 0 put b = x - m and c = 0. Then (2.13), {2.16) and {2.17) are clearly satisfied, and {2.14) and (2.15) become

r

(2.14')

and

K (y) + K ( - y) ¾ O,

O¾ y ¾ 1.

{2.15*)

Since the integrals in {2.12) are nonnegative and their sum is zero, it follows that each of them vanishes. Hence, since the first integral in (2.12) vanishes, it follows that u(x, y) = 0 in D. We have thus proved the uniqueness of a quasi-regular solution to the Frankl' problem, provided that conditions (2.14) and (2.15) are satisfied. §3. Existence of the solution of the Frankl' problem We shall prove the existence of a solution of the Frankl' problem in the case K(y) = sgnylylm (see (16b]). In this case equation (1.1) is (3.1) sgn YI Yim Uxx + Uyy = 0. Here A'C is a segment of the characteristic 2 X

+ m +2

m+2

( - y)

2

2

= m+2

·

We claim that there exists a function u(x, y) with the following properties: 1) It is a regular solution of equation (3.1) in D. 2) It is continuous in the closed domain D. 3) Its partial derivatives ux and uy are continuous in D, with the possible exception of the points 0(0, 0), A (O, 1), A' (0, -1), C(2/(m + 2), 0) and B(a, 0), where they may have poles of order less than unity. 4) It satisfies the boundary conditions (1.2)-(1.5). Let D* denote the domain symmetric to D with respect to the y-axis, and

§3. EXISTENCE OF THE SOLUTION

189

The domain DT is bounded by r + r* and the segment [-a, a] of the x-axis. where r* is the curve symmetric to r with respect to the y-axis.

D1 = D

+ D* + A'A.

Define the function 1/i'j(s) on length of r + r*.

r*

to be equal to 1/1 1 (/

-

s), where/ is the

We wish to find a solution of ( 3 .1) in D 1 , satisfying the boundary conditions

u Ir•= lj); (s),

u Ir= 'lj, 1 (s), u (x, 0)

= 'lj,2 (x),

u (x, 0)

=

lj,2 ( -

(3.2)

2

m+2 - (x) =

~i

f(

S~x - b2 ~

Xs) qi (s)ds,

(3.19)

0

We may thus reduce (3.17) to the following equation:

(1 - inJ...) ct>+ (x)- (1

+ inA) x),

(6.11)

tlx

(6.12) Consider the behavior of 3/1 (x, t)/3x as t --+ x. We shall find an estimate for the integral in (6.11) as t--+ x (t > x) and for the integral in (6.12) as t--+ X (t < X). For the first of these integrals, assuming t < 2x, we have

§6. EXISTENCE THEOREM

and for the second, if t

213

> x/2,

Therefore, from ( 6.11) and ( 6.12) we obtain an estimate valid for O < x

< 1: = 0 (In It -

a11 (x, t)

ax

x

I).

(6.13)

Hence, in order to differentiate the expression I

f / (x, t) v (t) dt

H (x) =

(6.14)

1

0

we can differentiate once under the integral sign:

f I

dH (x) dx

=

v (t) dt.

a1 1 (x, t)

ax

0

Differentiating (6.11) and (6.12) with respect to x, we get a211 (x, t)

ax 2

= (!...) 1- 2~ ~ • X

(6.15)

i-X

Hence it is clear that we cannot differentiate a second time under the integral sign in (6.14). Consider the expression x-&

j(x, e)

=

J a\~·

o

I

t)

v (t) dt

+

f

ar

1) ; ·

t)

v (t) dt.

x+e

Obviously,

. "( x, e) =-d. dH I1m1

e➔ O

X

Differentiating (6.16) with respect to x, and using (6.15), we get

(6.16)

V. THE MODIFIED TRICOMI PROBLEM

214

f + f (__!-_)1-213 2~ x-e

dj (x, e) -

dx

-

I

v (t) dt .

x

o

t-x

(6.17)

x+e

+v(x-e) aI 1 (x,t)I OX

-v(x+e)a/1(x,t)J t=x-e

l=x+B

OX

Let us evaluate the limit of the difference I= o/1

I

(x, t) OX t=x-e

-

o/1 (x, t)

I

OX

as

e-0.

l=x+e

By (6.11) and (6.12), we have

lim I= . 2n~ B➔ O

sm 2Jt~

+ lim [(~)-2/:l - (x + e )-213 e➔O

x

x

(6.18)

f µ-1-2/:l 00

=

+ 2~

21t~ sin2n~

dµ

I-µ

0

Obviously, the convergence as E --➔ 0 is uniform in x, 0 < x < 1. The integral on the right side is interpreted in the sense of the Cauchy principal value. It is readily shown that

f µ-l-2i3 00

dµ

1_ µ

=-

:n; cot 2:n;~.

0

Therefore, from (6.18), Jim [ 0/1 (x, l)J e➔O ox t=x-e

I

iJ/1 (x, t) i)x t=x+e

] = 2:rt~ tan na.

(6.19)

Now, add and subtract v(x - 1:)o/ 1 (x, t)/axlt=x+E on the right of (6.17). Then

+ [ a\~• t)

J

-

iJ/1 (x, t)

iJx

l=x-e

ol 1 (x,

+ ax Hence, by (6.19) and (6.13), the limit

I)

I

I ]. v (x - e) l=x+e

l=x+e

[v (x -

e) - v (x + e)].

§6. EXISTENCE THEOREM

215

(6.20) exists, and the convergence as e - 0 is uniform in x (0 < x < I). Then, by a standard theorem of classical analysis, the derivative d 2 H/dx 2 exists and 2H • --'-...:......:..---'dj (x, e) IIm = -d -

dx

t➔O

(O 2 (x) and its derivatives ;(x) and cI>;(x) as x -+ 0 and x -+ l. Note that if we replace x by (1 - x) and t by (1 - t) on the right of (6.29), the expression remains unchanged, except that the argument of~ is not t but (I - t). Thus 2 (x) displays the same behavior near the points x = 0 and x = 1, so we can confine ourselves to the limits of i(x) and its derivatives as x -+ 0.

§6. EXISTENCE THEOREM

217

After this remark, we differentiate ( 6 .29) twice with respect to x, and for brevity set Hkt

= k2

j

qi (t) /+v

(½- 11\1 -

t)v

(¾- 11) [x 2 + (1 -

2x) ,1-(l+IH) dt. ( 6.30)

0

Then we can write (6.29) and the derivatives in the form

), Hoo= o (xv-4-2tl o+v>), H 12 = o (xv-2-211 c1 +v>).

r-

~(l

Substituting these estimates into (6.31), we get

CI>2 (x)=O(x),

CI>~(x)=O(l),

~(x)=O(xv-•- 2IICt+v>).

(6.23)

We can now determine 'Y from the condition

v-1-2p(l +v)~o.

(6.33)

i.e.

(-1