Ledure Notes on
Mixed Type Partial Differential Equations John M Rassias Pedagogical Department The University of Athens 33, Ippocratous Str. Athens Greece
&h World Scientific . , Singapore. New Jersey • London • Hong Kong
Published by World Scientific Publishing Co Pte. Ltd POBox 128, Farrer Road, Singapore 9128 USA office 687 Hartwell Street, Teaneck, NJ 07666 UK office 73 Lynton Mea9, Totteridge, London N20 8DH
LECTURE NOTES ON MIXED TYPE PARTIAL DIFFERENTIAL EQUATIONS Copyright © 1990 by World Scientific Publishing Co Pte Ltd
All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher
ISBN 981-02-027S-X 98J-02-0406-X (pbk)
Printed in Singapore by JBW Printers & Binders Pte Ltd.
PREFACE This series of lecture notes includes various parts of the theory of mixed type partial differential equations with boundary conditions
such as: the classical dynamical equation of mixed type due to S. A. Chaplygin (1904), regularity of solutions in the sense of the first pioneer in the field F. G. Tricomi (1923) and in brief his fundamental idea leading to one-dimensional singular integral equations, the characteristic problem due to F. I. Frankl (1945), the mixed type equation due to A. V. Bitsadze and M. A. Lavrentjev (1950), the classical a, b, c, energy integral method for mixed type boundary value problems and quasi-regularity of solutions in the sense of M. H. Protter (1953), weak (or strong) solutions in the classical sense, well-posedness in the sense that there is at most one quasi-regular solution and a weak solution exists, a selection of new results, and open problems.
The present book is a revised and augmented version of a lecture course delivered by me at the I.C.M.S.C.-U.S.P., Brasil from August 11, 1988 to September 9, 1988. Deep gratitude is due to all those who have contributed to this work and have generously helped me carry out this project. My very special thanks to AIda Ventura for his great help and kindness while staying at the beautiful environment of the Instituto de CiEmcias Matematicas de Sao Carlos-U.S.P., Sao Paulo, Brasil. My special thanks to Antonio Fernandes Ize who along with Professor Ventura invited me here at I.C.M.S.C.-U.S.P. and to Luiz Antonio Favaro (Director of I.C.M.S.C.), to Hildebrando Munhoz Rodrigues (Head of the Mathematics Department of I.C.M.C.S.) and to the Foundations: Coordena<;ao de Aperfei<;oamento do Pessoal de Ensino Superior (C.A.P.E.S.) and Financiadora de Estudos e Projetos (FIN.E.P.), and to the Instituto de Ciencias Matematicas de v
vi
Preface
Sao Carlos (LC.M.S.C.), Universidade de Sao Paulo/Brasil for their support. Finally my appreciation to the typist, Luiz Carlos Franco, for his excellent job on my present work.
Sao Carlos, 1988
John Michael Rassias
CONTENTS
Preface
v
1. The dynamical equation of mixed type
1
2. The Tricomi problem
6
3. Regularity of solutions (in the sense of Tricomi)
8
4. Fundamental idea of Tricomi
9
5. The Bitsadze-Lavrentjev problem
33
6. The Gellerstedt problem
41
7. The Frankl problem
43
8. Quasi-regularity of solutions (in the sense of Protter)
54
9. The a, b, c, ... energy integral method
55
10. Weak (or strong) solutions in the classical sense
86
11. Well-posed ness
123
12. Open problems
137
13. Basic Books
138
14. Subject Index
139
15. Author Index
143
vii
1.
A Gas Dynamical Equation
of Mixed Typ e
See: s. A. Chaplygin (ON GAS JETS, Scientific Annals of the Imperial University of Moscow, Publication no. 21, 1904. Translation: Brown University, Providence, R.I., 1944), M. H. Protter (J. Rat. Mech. Anal., 2, 1953,721-732), and J. M. Rassias (MATHEMATICS & SPACE TECHNOLOGY, Athens, 1981). Consider a two-dimensional adiabatic potential flow of a perfect gas. The stream function ,p = ,p(x, y) satisfies the equation
L1/J = (la 2 -1/J~),pxx + 21/Jx1/Jy1/Jxy + (p 2 a2
-
,p;)1/Jyy
(1)
=0,
where
= the local velocity of sound, p : = the density of the gas.
a :
Equation (1) is transformed to a linear equation of mixed type by 1
applying the hodograph transformation:
(2) where tt, V :
the rectangular velocity components as new independent variables.
The corresponding components in polar coordinates are:
0= tan -1 (v) -.
(3)
tt
To normalize r introduce
(4) which is dimensionless quantity, as new independent variable, where ro : the speed corresponding to zero density.
Therefore equation (1) becomes
a {2t
at
(1 - t)P,pt
}
+
1 - (2,8 + 1 )t
2(1 _ t)P+1 ,p66 =
a,
(5)
where
Cp :
the specific heat at constant pressure,
Cv :
the specific heat at constant volume.
Note
(6)
Lecture Note. on Ml%ed 7Upe Partial DiJTerenti(J.1 Equation.
3
where
Po : the density of gas at zero speed,
k : satisfies the weU-known relation (: = const.) p
= kp"l
(7)
Introduce new independent variables ~
=
(J ,
YJ = -
j
(I -
t
u)~
2u
-L\IJ!+f
du .
(8)
Then the equation (5) takes the form
1 - {2,8 + l)t
(I _ t)2~+l ,pee
Set by K = K (YJ) the coefficient of Note
+ ,p.,.,
=
a.
(9)
,pe e·
{3 : is a positive constant (~ 2.5 for air) , "y:
is a positive constant
(~
1.4 for air) .
Besides
K{O) =
a
(10)
because YJ
=a
for
t =
1/ (2,8 + 1) .
This case corresponds to points where the velocity is equal to the local velocity of sound, and therefore (9) is parabolic. Moreover
(11)
4
J M Rauia,
because '1
> 0 for
t
< 1/(2{3 + 1)
corresponding to subsonic velocities, and (9) is elliptic. Finally
because '1
< 0 for
t
> 1/(2{3 + 1)
corresponding to supersonic velocities, and (9) is hyperbolic. Therefore equation (9) is of mixed type.
Remarks: i. The velocity potential ¢> = ¢>(x, y) and the stream function 1/; = 1/;(x, y) satisfy Cauchy-Rt'emann equations (12)
ii. The discrimt'nant of equation (1) is given by the formula
Then
(14) where M : Mach number: r =(1/;;
r/a,
+ 1/;~)t /p .
iii. A flow is called subsonic, sonic or supersonic at a point as the flow speed r is: < a, = a, or > a, respectively. These three cases correspond to: [) > 0, = 0, or < O.
Lectu.re Note, on MIXed Type Partial Differenti.al Equation,
5
iv Transonic flows involve a transition from the subsonic to the supersonic region through the sonic. Therefore transonic flows are the most interesting. v. Transition from subsonic to supersonic flow becomes possible: Two sections of cones or similarly shaped tubes with the same axis are placed opposite each other and connected, thus forming a de Laval nozzle with "entry section" , "throat" , and "exhaust section". Then a subsonic expanding flow in the entry section, on passing via the throat, can change into a supersonic expand-
ing flow in the exhaust section. vi. Equation (1) is quasi-linear and is converted to the linear equation of mixed type (9). The corresponding equation to (1) is the quasi-linear equation
Lr/>
= (a 2 = o.
-
r/>;)r/>xx - 2r/>xr/>yr/>xy
+ (a 2
-
r/>~)r/>yy
(15)
where r/> = r/>(x, y) is the velocity potential. vii. Equation (15) comes from the Euler continuity equation
Lr/>
= (Pr/>x)x + (pr/>y)y = o.
(16)
6
J. M. Ra"ia,
2.
The Tricomi Problem
In 1923 F. G. Tricomi (Atti Accad. Naz. Lincei, 14, 1923, 133-247) initiated the work on boundary value problems for partial differential equations of mixed type and related equations of variable type. The Tricomi Problem or Problem T: consists in finding a func-
tion u
= u(x, y) which satisfies equation (the Tricomi equation)
in a mixed domain D which is simply connected and bounded by a Jordan (non-selfintersecting) "elliptic" arc 91 (for y > 0) with endpoints 0 = (0, 0) and A = (1,0) and by the "real" characteristics :x
92
2
3
+ - (- y).- =
93 : x -
3
2
3
a( -yF
1 ,
=0
of (*) satisfying the characteristic equation
y(dy)2
+ (dX)2 = 0
(17)
such that these characteristics meet at a point P (for y < 0), and assumes prescribed continuous boundary values {
u
= ¢(8)
U
= 1jJ(x)
(**)
Denote
D1
= Dn{y > o} : elliptic region, D2 = Dn{y < o} : hyperbolic
region OA := D n {y = O} : parabolic line of degeneracy. Consider the normal curve of Tricomi with equation 2
o
91 :
(
x -
1 4 )
2 + gy 3 = 41
Lecture Note. on MlZed Type Partial Differential Equation.
y
x
p Fig
or equivalently
g~ where
:
1
Iz - 2\I' = 21 '
.2
L
z=x+'"3 Y "
.
r-;
,=y-l,
such that g1 contains g~ in its interior.
7
8
J M.
Ra"ia~
3.
Regularity of Solutions
Definition 1. A function u = u( x, y) is a regular solution of Problem T in the sense of Tricomi if: 1) u is continuous in jj : closure of D(:= D u aD), aD = gl U 92 U g3 , are continuous in fJ (except possibly points 0, A, where they may have poles of order: < ~, i.e. may go to infinity of order: < ~ as x ~ a and x ~ 1, i.e. U.z = O(r-t + e),u y = O( r- t + E), where r := distance from 0 or A), 3) U.z.z, U yy are continuous in D (except possibly on OA where they may not exist), 4) u satisfies equation (*) at all points of D\OA (i.e. D except OA), 5) u satisfies boundary conditions (**). 2)
u.z, U.z
Lecture Notu on Miud Type Parbal Differential Equation.
4.
9
Fundamental Idea of Tricomi
The fundamental idea of Tricomi of finding regular solutions for Problem Twas
First:
to solve Problem N (in Dd: To find a regular solution of equatio n (*) satisfying the boundary conditions u = ¢>
{
uy = v
Second:
on 91 on OA
(N)
where v = v(x) is continuous for x : 0 < x < 1 and may go to infinity of order: < 2/3 as x ---+ 0 and x ---+ 1. to solve Cauchy- GouTsat Problem (in D 2 ) treating
v(x) = uy (x, 0) as a known function, i.e., First to solve the Cauchy Probl em (in D 2 ) of equation (*) (in D 2 ) satisfying the conditions
= r
on OA
uy = v
on OA
u
{
(C)
where r = r(x) is continuous for x : 0 < x < 1 and may go to infinity or order: < 2/3 as x ---+ 0 and x ---+ 1, and Second to take into account the boundary condition u
= 1/J
on 93 ;
10
J M Rauia8
therefore we have the GoursatProblem (in D 2 ) of equation (*) (D2) satisfying the boundary conditions {
u=r
on OA
u=t/J
(G)
Denote
e= x -
2
}.
-( -y) 2 3 2 3 '1=x+-(-y)2 3
(characteristic coordinates)
Then g2
'1 = 1
g3
e= a .
Let gl
:
x = x( s) , y = y( s),
(parametric equations of gl)
(18) where s : arc length reckoned from A . Assume the following conditions on gl : i) The functions x = x( s) and y = y( s) have continuous derivatives x' (s) and y' (s) which do not vanish simultaneously on [0, I], 1 := length of gl, and the second derivatives x" (8) and y" (s) satisfy a Holder condition on [0, I],
Lecture Note. on Mazed TlIPe Partial Differential
~quatioru
11
ii) In neighborhoods of the points 0 and A on 91, (19) where C
constant.
Reduction of Cauchy Problem (in characteristic coordinates ): The Cauchy problem (*) and (C) is equivalent to the following problem in characteristic coordinates:
Eu = ue" = 0
+ A(~,'1)ue + B(~,'1)u" (Euler-Darboux equation)
and
where
1
1
A=A(~''1)=6'1_E ' B
1 1 = B(E, '1) = -6'1 ---~ .
Solution of Cauchy Problem
u(E, '1) =
[*l
and
[el
Ie" 9(E, '1; t)r(t)dt - Ie" h(e, '1)lI(t)dt
(cf. K. 1. Babenko: Doctoral Dissertation, Steklov Inst. Math., Moscow, 1952),
(20)
where
where
Important Condition: Consider triangle OAP bounded by i) the segment OP of the characteristic 93 : E= 0, ii) the segment PA of the characteristic 92 : '1 = 1, iii) an arc of the parabolic curve OA : '1 = E i.e. ~OAP := {E,'1 E [0,1],'1 >
o·
Reduction of Goursat Problem (in characteristic coordinates) : The Goursat problem (*) and (G) is equivalent to the following problem in characteristic coordinates: Equation (*) and
[G] Solution of Goursat Problem l*l and [G] (Special Case: 1/J = 0): To solve the Goursat problem in this case we assume: u = u( E, '1) to be a twice differentiable function of equation in the triangle ~OAP satisfying the boundary conditions [G]. Then
l*l
(21 )
Lecture Note. on Miud Type Partial Differential. Equatiom
satisfying equation
l*l
13
and boundary conditions [G]( 1/; = 0).
In General (and combining Cauchy & Goursat Problem): It is known G. H. Hardy and J. E. Littlewood (Math. Z. 27 (1927/28), 565-606) that: If /(x) satisfies a Holder condition on (0, 1) with exponent a, it may be expressed as
/(x) = /(0)
+
1 x
(x - t)P-l g(t)dt ,
(22)
where
9 = g(t) satisfies a Holder condition with exponent
a - (3
> O.
Important Conditions
1) r = r(t) satisfies a Holder condition with exponent al > ~ for o ~ t < 1, 2) II = lI(t) satisfies a Holder condition with exponent a2 > ~ for o ~ t < 1. Therefore r = r (t) and II = II (t) may be expressed as
where
> 0 is sufficiently small, and 4> = 4>( s), 1/;( s) : are continuous functions for 0 ~ s < 1. g
Substituting (23) into (20), changing the order of integration and using the integral representation of the hypergeometric function
14
J. M. Ra"ia,
F we get
ti(~, '1) =
Ie
(!6 )6 - e)3 ; '1'1-- ~) ds + Ie'! cP2 (s) ('1 - ~) - t ('1 - s)· (~ ~ 1 + ~ =;) ds o
cPds)('1 - s)- t+. F
8
F
1
+ e cP3 (s )( '1 x F
(! + ! 6
+ r(O) -
e
'6'
,
,
e;
s) - t (~ - s)·
(24 )
~ - s) ds ''1-S
1+e .
(~) t ('1 - ~)tv(O) ,
where
The Cauchy-Goursat Problem The Cauchy- Goursat Problem: consists in finding a solution of equation (*) in D2 (= D2
u aD2 ) and satisfying boundary { 1.£le=o = 1/;('1) 1.£,,1,,=0 = v(x) .
conditions
(CG)
The Tricomi Problem The Tricomi Problem: consists in finding a solution of equation
(*) in D assuming prescribed values on
91 and on the characteristic
93:
(T)
Lecture Note! on Mixed Type Partial Differeno'a/. EquatioJU
15
where
4>( 8) : satisfies a Holder condition with exponent
o!,
and
,p('I) : has a bounded first derivative satisfying a Holder condition with exponent (:J.
The Hypergeometric Function The integral representation of the hypergeometric function F is
F(a b c· II) =
, ",..
r(c) t t a - 1 (1_ t)c-a-l(l_ ,IIt)-bdt r(a)r(c - a) 10 .., 0< Re(a) < Re(c) . (25)
Integrals of Fractional Order The expression 1
fa (x) = r(O!)
r (x -' -1t-
10
1
f(t)dt ,
O~x
(26)
f = f(t) : integrable on (0, 1) , 0<>0 is called an integral of fractional order a (compare (26) with (22)). We often use expressions of type (26).
Solution of Cauchy Problem (*) and (C)
u(x, y) = /1
1 1
r[x
o
r (§.)
+ f3
2
+ -( -y)
t
1. 2
3
t~) y loll [x +
.l
(2t - 1)][t(1 - t)r 6 dt
(27)
2
3(- y)
1. 2
(2 t - 1)][t (1 - t) r
1. 6
dt
Equating expression (27) on characteristic 93 to function ,p{ x) (i.e. combining with Goursat Problem) and applying the well known inversion formulas
F{x) = sin(7rO!) ..!!:..-1'" 7r
dx
0
(t) (x - t)1-a
dt
(28)
16
J. M. Raaaia,
to the Abel integral equation
I ,.
F{t)
o (x -
t)a
dt
=
0 < a < 1 ,
(29)
we get the First Fundamental Functional Relation for Problem T:
r{x) = 1/;1 (x)
+
il" o
lI{t) 1. dt , (X-t)3
(30)
where
Solution of Problem N By direct checking we see that
I~
- xl- t : is a
solution of (*) ,
where ~
c = ..
2 +,. X -'1' 3 3
(~
i= x,
'1 < 0) .
Denote
G=
G{x;~, '1):
the Green's function
i.e. the function of the form
G{x;~, '1) = I~
- xl-} + Go{x;~, '1)
,
where Go = Go{x;~,
'1) : is a solution of (*)
(31)
Lecture Note! on Mized Tllpe Partial Differential equatiol'U
regular with respect to
€,
17
and '7 in D 1 , and satisfying the conditions
{a -
~~~I-I~ xl- t ,~ E g1
(32)
=0.
'7
'1=0
In the special case when 91 coincides with the normal curve go1 .•
(€ - 21)2 +
4 9'73
1
=4
or equivalently
g~
:
I~
1
1
- 21 = 2 '
the function (33) and therefore from (31) the function.
(34) Isolating the point (x,O) by the curve g~ : I~
g~
of the normal contour
- xl = c
from domain D1 and applying Green's formula (to the remaining part of Dd:
J (~~ '7
- G
~~) d'7 -
(u ~~ - G ~~) d{ =
0
we get (as c ~ 0) from (32) that
(35)
J. M Ra66ia6
18
where N : is the inner normal of the boundary aD,
ae aN
ae a~ ae aT] := T] [if aN + aT] aN :
is the co-normal derivative associated with Tricomi operator. But
Therefore from condition (N) we have the Second Fundamental Functional Relation for Problem T:
r(x)
+I
11
e(x; c, O)//(c)dc
=
r (x) ,
(37)
where
Special Case (when gl coincides with
g~):
The expression (37) becomes simpler:
r(x)
+I
t [
10
1
1
It-xli
-
1
1 ]
(t+x-2tx)i
//(t)dt = F* (x),
where F* = F* (x) is analytic function of x for
a<
x < 1.
(38)
Lecture Notu on Mazed Type Partial Differential Equatioru
19
Eliminating r = r(x) from functional relations (30) and (38) we get
1'"
lI(t) ---'-'--:-1. dt o (X-t)3
+
11 [' 1
1.
1]
-
Ix-tl3
0
1.
(t+X-2tX)3
- cPl
lI(t)dt -
*
(x) , (39)
where
cP~
!
=
[F* (x) - ,pI (x) 1 '
"1
Applying the inversion formula (28) to the Abel integral equation (29) we write (39), as follows:
II(X)
V3 d +211'dx
1
11
lI(t)dt
0
x
o
=
[1oIE-tI3(X-E)3 X
dE
1.
de..
]
1.
(40)
(E+t-2tE)t(x-E)t
3
2F (x) ,
where
F(x) = _1_ ~ lI'V3dx
1 x
0
cP~ (t) 1. dt . (X-t)3
When x lies strictly inside the interval (0, 1) the second integral term on the left side of (40) can be written in the form
(41)
3
=
1
-"2 11'V3
11 (t) 0
:;;
lI(t)
t t
+ x - 2tx dt ,
20
J. M. Ra88ia8
where
Denote
I(x) =
1 I'" 1",-· [It + J'" 1 I'" lI(t)dt
1
o
I. (x) =
lI(t)dt
o
d€
1
d€ 0 (t - ~). (x - €) • d€ ]
1..1
1.
.1
(€-t).(x-€).
t
+
2
I€-tl'(x-€),
0
1
lI(t)dt
"'+ <
0
d€
1.
.1
(t - €). (x - €).
Then
I(x) = lim I. (x) .
(42)
€ = t + (x - t)z
(43)
• ~O
Using the substitution
we get
J'" t
d€ ---=-----::= (€ -
t)}
(x - €) f
11
l
z-1. · (1 - z )_.1• dz
0
(44) x> t .
Using the substitution
x-€
Z3= - -
t -
€
we get
x> t
l
Lecture Note. on Mixed TWe Partial Differential Equatio....
21
or
x>t
(45) where
(-X)t t
~-
.
Besides
r 10
fO
dE
(t-E)t(x-E)~
=-3
dx l-z3
~
+ inVI + ~ + ~2 In 1 (2~ + 11" + y3tan-- --
= -in(1
- ~)
y3
1)
y3'
x < t .
(46) Then substituting (44)-(46) into the integral expression of I. (x) above we get
I. (x) =
11"In [
r-· v(t)dt -11
2y3 10
V(t)dt]
x+c
+ ([-. +
l:J [
+ V3tan-'
C(Ja!) ]V(t)dt .
-Inl! -
(I + In";! + (+ ('
(47)
Differentiating I. (x) of (47) with respect to x we get
I; =
11"
In
2y 3
[v(x -
e) + v(x + ell +
+ v(x - e) [ - in(~ -
(l 0
1)
X -"
+
11) x+.
+ inVI + ~ + ~2
(t)} v(t) - ~dt x
t
x
22
J. M Ra"ial
+vatan-'
C'X)] <~(*')t
+ V(X + e) [in(1 - ~) - in';l + ~ + ~2 - va tan -. Letting e
C',,; 1) ]
(48)
•
r=(;tc)3
0 we get from (42) and (48) that
---t
.-+ I; (x )
l' ( x) = lim
0
11" ..J3
= -v(x)
+
11 0
(t) - t -v(t) -dt x
(49)
t-x
Therefore we find
..J3 ~ 211" dx
11
v(t)dt
0
IX 0
d~ I~ - tl t (x -
~v(x) + ~_1_11
=
2 11"..J3
2
0
(!) t x
~) t
(50)
v(t) dt . t - x
From (40), (41) and (50) we get
v(x)
+
1 11 ( 1 to
1I"y 3
-
0
t- x
-
1)
t + x - 2tx
v(t)dt = F(x)
(51)
which is a one-dimensional singular integral equation with respect to v for Problem T. This equation (51) is equivalent to Problem T in the case when gl coincides with the normal curve g~ (the latter curve gives a simple Green's function).
General Case (When gl does not coincide with gn: Eliminating the function r = r(x) from expressions (30) and (37) as above we get the general one-dimensional singular integral equation with respect to v for Problem T
Lecture Note. on Mi:r.ed Type Partial Differential EquationJI
23
which is:
II(X)
+ +
1 11 (t)X t ( 1X
r.; 1ry 3
11
-.
0
t -
-
t
1)
+ X-
2tx
lI(t)dt (52)
K(x, t)lI(t)dt = F(x) ,
where
K = K(x,t) : is expressed through Go(x;t,O) .
Tricomi'8 Theorem: If F{x) satisfies a Ho'ider condition for a < X < 1 and F{x) P E L (0, l),p > 1, then the solution of the equation (51) in the class II(X) such that xt II(X) E L P (0,1) is given by
II(X) =
3 {
4 x
1 F(x) - 1ry'3
C~
11 [ 0
t(1 - t) ] t x(1 - x) (53)
x - t
+ x 1_ 2tx)
F(t)dt}
where A is an arbitrary constant.
Remarks: 1) Since II = II (x) may have a pole of order less than ~ as x --? 0 we get A = 0 in (53). 2) With the help of the above Tricomi's theorem the singular equation (52) is regularized. Therefore we have Regularization of the Singular Equation (52). Lemma 1. 1/ t/> = t/>( s) satisfies a Holder condition with exponent a, and {
+ t/>1 (s) t/>(O) + t/>2(S)
t/>{s)
= t/>(l)
t/>(s)
=
,
(54)
where l: length of g1
(55)
< C's
and 1/;' ('1) satifies a Holder condition with exponent (3, then F(x) satisfies a Holder condition with exponent + E, E > 0, for
*
O:O:;x
Proof of Tricomi '8 Theorem Set ¢(x) = xtv(x) , { f(x) = xt F(x) .
(56)
Then singular integral equation (51) is reduced to ¢(x)
+
1r.; 'lry
3
11 ( -
1
t - x
0
-
x
1 ) ¢(t)dt = f(x) .
+t -
2tx
(57)
Let z be an arbitrary point of the complex plane. Following Carleman (Ark. Mat. Astr. Fys. 16, No. 26, 1922), we set ell(z) = -1.
2'1rl
11 (
- 1
t - z
0
-
z
+t
1 ) ¢(t)dt -
2tz
(58)
It is clear that ell = ell ( z) : is holomorphic in both upper
and lower half-planes, and for 0 < x < 1 : ell+ (x) - ell- (x) { ell+ (x)
+ ell- (x)
= ¢(x) =
~ 'lrl
11 (_1_ _ + 1 ) 0
t- x
x
t - 2tx
Therefore (57) is reduced via (59) to the equation
¢(t)dt. (59)
Lecture Note! on Mized TlIPe Partial Differential Equatioru
25
Then from (58) we get
cI>
(_z_) = (2z - 1)cI>(z) . 2z - 1
(61)
Note: that the Mobius trans/ormation: z 2z - 1
(62)
~=--
maps the upper half-plane onto the lower, and conversely; the interval (0, 1) is mapped onto the two rays (0, 00) and (-00,1). Replacing x by
(1+
~) y'3
2%%_1
into (60) and using (61), we get
cI>- (x) - (1 -
~) y'3
cI>+ (x) = _1_/ (_x_) , 2x - 1 2x - 1 -oo<x
Denote
~1
1-
G(x)
iy'3 2
1 + iy'3 2
0< x < 1
-00 < x < 0 ,
1<x<00,
y'3 y'3 + /(x) , 0<x<1 H(x) =
_
J3 J3 - i
_1_ x / 2x - 1
-00 < x < 0,
(_X_) 2x - 1
1 < x < 00 ,
Then combining (60) and (63) we get the equation
cI> + (x) - G ( x ) cI> - (x) = H ( x),
- 00 < x < 00 ,x =I- 0 ,x =I- 1 . (64 )
J. M. Ra"ia8
26
Therefore the solution of the singular integral equation (57) is reduced to the following
Problem of Complex Analysis: Find a function = (z): holomorphic in both upper and lower half-planes and satisfying the boundary condition (64).
Solution: The solution of this problem can be obtained in explicit form. In fact, the solution
X(z)
= exp
11
[ --1 6
(1- 2z - 1 ) dt ] t- z t + z - 2tz
0
(65)
is a particular solution of the corresponding homogeneous problem: (66) and satisfies the condition
X(2z~ 1)
(67)
=X(z).
From (65) we have
J
=e .::.iL 6
X+(x)
1
=e
X- (x)
ll.( 6
(
X
--
)}
I - x
x
--
I- x
'
O<x
)t
'
O<x
and, by (67) and (68),
X+ (x) { X-(x)
!L(
=e 6
x I - x
--
)t
X )} 6 =e =.!L( --
I-x
-00
' ,
<x<0)
-00
<x <0 ,
l<x
(69)
Lecture Notu on Mized Type Partial Differential Equatioru
27
Replacing G(x) by X+ (x)j X- (x) in (64) and dividing by X+ (x), we get: tI>+ (x)
tI>- (x) _
X+(x)
X- (x)
H(x) X+ (x)
(70)
One particular solution of this equation is tI>(z) = X(z)
21ri
foo
H(t) ~ X+ (t) t - z
-00
(71)
To find the general solution, we consider the homogeneous equa-
tion _tI>_+-'---(x-=-) _ tI> - (x) _ 0
X+ (x)
X- (x) -
.
(72)
Note: This equation (72) shows that the function
w(z)
=
tI>(z)
(73)
X(z)
is holomorphic in the whole plane, with the possible exception of z
=0
and z
= 1, which can
only be poles. Besides,
w(z )vanishes at
infinity.
Assuming that tI>(z) may have poles of order less than 1 as z --. 0 or z --. 1, we see that
w(z)
1
= a-z
,
where a : = arbitrary constant. Thus the general solution of equation (64) is: tI>(z) = X(z)
21ri
1
00
-00
=X(~)[/O 21r1
-00
H(t) ~ + a X(z) X+(t) t - z z
+t+/OO]. 10 1
(74)
J. M. Ra"ia8
28
In the integrals
f
and
o
2t ~ 1.
z =
X (z)
()
211"i
00
1
-00
above we replace t by
1
Then from (67) we get:
t va JJ!L (_1_ _ 1 ) dt + a X (z) 10 va + i x+ (t) t - z t + z - 2zt z
.
(75) Define
4>(x) = +(x) - -(x) ,
0 < x < 1 .
(76)
Then from (75)-(76) we get
3[ 1I"va 1 (- t 11 (1- t) t )f (t] )dt + Ax ( 1x + x1 - 2tx x ) 1- x
4>(x) = - f(x) - 4
X
-
-
t -
t
0
(77) .l 3
t
(1 - x) -
l. 3
,
where A : = arbitrary constant. Returning to the previous functions we get (53); thus completing the proof of Tricomi's theorem.
Remarks:
I.
The integral
1x -
t(t)t( t 10;;
t
1)
+ x - 2tx v(t)dt
one-dimensional singular integral, i.e. it is the principal value of a divergent integral. Singular integral equations were studied for the first time by T. Carleman (1922) IS
Lectu.re Notea on Mized Tllpe Partial Differential Equ.atiofU
29
(Ark. Mat. Astr. Fys. 16, No. 26, 1922). The kernel of the integral equation (51) contains the addend: = t+Z~2tz which is neither Cauchy' kernel nor summable. Therefore equation (51) does not belong to the class of Carleman's equations. To solve (51) F. G. Tricomi thought as follows: Denote the kernel
L(t,x):=
(!.-)x t (_1 __1_) t - x t + x - 2tx
(78)
and kernels Ln+ 1 (t, x) :
Ll (t, x)
=
L(t, x) 1
= -n
n-l
L
Li+l(t,w)L n- i (w,x)dw,
(79)
i=O
n ~ 1.
If L(t, x) were an ordinary Fredholm kernel, then all the addenda in (79) would be equal and we would have the ordinary iterated kernels. But these addenda are different in Tricomi's
case because: the ordinary formula of integration order
f ails for a repeated singular integral. Tricomi has shown that 1 [(l-t)X]n Ln+l(t,X)=-L(t,x) 2ln( ) ,
n!
t 1- t
which is valid for the kernel L of (78).
n~l
(80)
30
J M Rauia6
II.
III.
The two solutions (one of Problem N in Dl and another of the Cauchy-Goursat Problem in D 2 ) obtained are then matched, together with their first derivatives on ~A. Let the function A(t): be continuous on [a, b] and have a continuous first derivative [a, b]' which may have poles of order: < 1 at the endpoints of [a, point of [a, b]. Then
j
a
b
b]' and let c: be an interior
A(t) dt = A(b)ln(b _ c) - A(a)ln{c - a) t -
C
-lb
(81)
A'(t)lnlt - cldt
Under the same conditions, we have
j
a
b
A(t) dt t-c
[_jC (c-tp-· A(t) dt A(t) d 1 + j (t-c)1-a
= lim
.~O
a
(82)
b
t
C
by using (81) and recalling that x· - 1 lim - - = lnx .~O e
Lemma 2. If
[00
T {a,,8; x) = 1
10
[00 T {a,,8;x) = 2
10
ta -
(83)
1
[x + (I _ x)t]P{I- t) dt , ta-1
[x+(I-x)t]P{I+t)dt,
where
a,,8 : constants : 0 < a < ,8 + 1,
a > ,8 ,
Lecture Notu on Mixed Type Partial Differential Equatiom
31
then
= 11" cot(a -
(3)11" +
Xu-f1
r(a)r((3 - a) F r((3) 1
,
(84)
= (1 _ x)
-(3r(a)r((3 - a + 1) F r(I+(3) 2,
where
FI , F2 : hypergeometric functions : FI -= F(a, a - (3, a - (3 + 1; x) ,
F2
= F (1 -
2X)
a + (3, (3, 1 + (3; 1 -
1- x
Lemma 3. If
J(x)
t
[t(1 - t) ] t (
= 10
xU - x)
1
t- x - x
1)
+t _
2tx
dt
then
J (x)
= _ ~ ~ r (k) r (y'3+3 2 r2
(k)
+srn) F
r(~) (1 5
n
F
(~ ~ ~. x) 3'3'3'
2X)
8 1-
_~
3'3'3;~ (I-x) •.
Proof of Lemma 3: Replacing t by the new variable x 1- t
C=tl-x we obtain
J(x)
=
1 +1 00
Et 5
[x+ (1- x)EJ'(I- E)
o
00
o
dE
cJ. 1,,'
5
[x + (1- x)EJ'(1 + E)
dE
(85)
32
J. M. Ra"ia.
Applying Lemma 2 we get
J(x) =T
1
(~,~;x)
completing the proof of Lemma 3.
+T2
(~,~;x)
,
(86)
Lecture Note. on M.ud TWe Partial Differenf.1·al Equatioru
5.
33
The Bitsadze-Lavrentjev Problem
In 1950 A.V. Bitsadze and M.A. Lavrentjev (Dokl. Akad. Nauk S.S.S.R., 70, 1950, 373-376) initiated the work on boundary value problems for partial differential equations of mixed type with discontinuous coefficients.
The Bitsadze-Lavrentjev Problem or Problem BL: Consists in finding a function 1.1. = tion
S9n(y)
1.I. xx
+ 1.I. yy
= 0
1.I.(x,
y) which satisfies equa-
(the Bitsadze-Lavrentjev equation)
in a mixed domain D which is simply connected and bounded by a Jordan (non-selfintersecting) "elliptic" arc 91 (for y > 0) with endpoints 0 = (0,0) and A = (1,0) and by the "real" characteristics 92
X -
93
X
Y= 1
+Y= 0
of (BL) satisfying the characteristic equation
_(dy)2
+ (dX)2 = 0
such that these characteristics meet at a point P (for y < 0), and assumes prescribed continuous boundary values of the (**) form Consider the normal curve of Bitsadze-Lavrentjev with equation
9~:
( -"21)2 + y2 x
=
41
(upper semi-circle)
or equivalently 1
1
Iz - 21 = 2 '
34
J M. Rauia8
y
x
p
Fig 2
where z = x
+ iy,
such that g1 contains g~ in its interior.
Definition 2. A function u = u(x, y) is a regular solution of Problem BL if it satisfies conditions 1), 3) and 5) of Definition 1 and besides if it satisfies the following two conditions: 2) u"" u y are continuous in jj (except possibly points 0, A, where they may have poles of order: < 1 i.e. may go to infinity of order: < 1 as x --+ 0 and x --+ 1), 4) u satisfies equation (BL) at all points of D\OA. Idea of Bitsadze and Lavrentjev The idea of Bitsadze and Lavrentjev of finding solutions for Problem BL was almost similar to the Fundamental Idea of Tricomi.
Lecture Notes on Mixed Type Partial Differential Equatioru
35
We are going to go through the main steps: Assume in both cases (i.e. Tricomi's or T-case and BitzadzeLavrentjev's or BL-case):
u{O)
= u{A) = a
(87)
The general solution of Problem BL in D2 (i.e.: -u""" +u yy is given by the familiar formula of D'Alembert:
= 0) (88)
where
fl
a~ 1/J
fd t ) , h
=
t
~
= h{t) : are arbitrary continuous functions on
1 twice continuously differentiable on
a<
t < 1 .
The general solution of equation (BL) satisfying condition: u on g3 is in D 2 :
=
(89) Therefore au _ au
ax
ay
= 1/J' (~), 2
y
= 0, a <
x < 1,
(90)
or equivalently
r' (x) - v( x)
= 1/J' (~), a <
x<1,
or equivalently
The First Fundamental Functional Relation for Problem BL:
r' (x) - v{ x) = 1/J' (~),
a< x < 1 .
(91)
36
J. M Rauia,
The Second Fundamental Functional Relation for Problem BL: Consider the fundamental solution: lnl~ -
zl,
~=
e+ i'1,
of equation (BL) in D1 (i.e.: Denote
G = G(x;
e, '1)
z = x"+ iy,
+ U yy
U xx
'1 > 0,
y>0
(92)
= 0).
: the solution of equation (B1) in D1
with logarithmic singularity at point ~=x,
O<x
(93)
G= -lnk-xl+Go(x;e,'1), where a regular harmonic function of
e,'1
in D1 satisfying the conditions:
G o(x;C'1)-lnk-xl=O, ~ E g1,
aaGo'1 I
(94)
= O.
'1=0
Note: The function G is the harmonic Green's function with singularity at ~ = x in D~ = D1 U D~ u OA, where
D; : is a domain which is the image of D1 in OA .
Liapunov Condition: Assume that curve g1 satisfies the Liapunov condition: i.e. the tangent of the angle formed by the tangent line of 91 and a constant direction (Jor example: the positive direction of the Ox-axis) satisfies Holder condition.
Lecture Notea on Mszed TlIPe Partial Differential Equatioru
37
Isolate point (x,O) from domain Dl by the curve'9~ of the semicircle: 9~ : IS" -
xl
= e,
11 ~ 0
and apply in domain D2 Green's formula
J( au
G aN -
11.
aG) ds =
aN
(95)
0 ,
where s: arc length of curve is measured from A in the positive (counterclockwise) direction, N: inner normal of the boundary aD 2 , and ;~ = !~ Taking into account the boundary condition 11.
= if>
Ifi + !~~.
on 91
(96)
we shall have from (94) and (96)
Hence, in the limit, e
-->
0, we get
The Second Fundamental Functional Relation for Problem BL:
rex)
111
+-
11'
[Go(x; E,0)
-lnlE - xllv(E)dE = if>.(x) ,
(98)
0
where
(99)
Special Case (when 91 coincides with 9n:
38
J. M Rauia.
The expression (98) becomes a simpler, as follows:
r(x)
111
+-
1r
[in(t
+x-
2tx) - inlt - xl]v(t)dt = ¢. (x) .
(100)
0
Eliminating r = r(x) from fundamental relations (91) to (100) we get:
v(x)
+ ..!:. dd
x
1r
where
10t
[in(t
+x-
2tx) - inlt - xl]v(t)dt = F(x) , (101)
F(x) = ¢~ (x) - 2 d: ,p (~)
.
Assume that x lies strictly inside the interval (0, 1). It is easy to see that
d -d x
11
in(t + x - 2tx)v(t)dt =
0
11 0
1 - 2t
t
+x -
2tx
v(t)dt .
(102)
But lim I.(x) = lim [ r.---0 .---0 10 = I(x) =
in(x - t)v(t)dt +
6
11
t
lx+
in(t - X)V(t)dt] 6
inlt - xlv(t)dt ,
where the limit exists uniformly in x. It is obvious that the uniform limit lim I; (x) = lim[v(x - e) - v(x + e)]lne
.-+0
.-+0
. (l
X
- hm
6---0
= _
t
10
-
6
0
11
v(t) v(t) ) --dt + --dt t-x x+.t-x
v(t) dt t- x
exists, where the integral is (in the sense of Cauchy): the principal value.
Lecture Note. on Mized TlIPe Partial Differential Equation.
39
A Well-known Formula d 11 lnlt - xlll(t)dt = - 11 -lI(t) -d -dt. x 0 0 t-x
(105)
From expressions (102) and (105) equation (101) takes the form
lI(x)
(1 + -11"111 -t - x + t +1x- - 2t2tx ) 0
lI(t)dt = F(x)
(106)
which is one-dimensional singular integral equation with respect to 1I for Problem T. This equation (106) is equivalent to Problem BL in the case when gl coincides with the normal curve g~ (the latter curve gives a simple Green's function).
Note:
1 t - x
1 - 2t
+ t + x - 2tx
t (
=;
1 t- x - t
1)
+ x - 2tx
(compare (51) with (106)). Similarly as in the Tricomi's case: expression (53) (Tricomi's Theorem) we get here explicitly also that
lI(x) =
!
2
{F(X) _
~ 11 11"
0
t) + t +1x- - 22tx
[xu- t)]} t(l - x)
(_1x t -
(107)
F(t)dt } .
General Case (when gl does not coincide with gn: Eliminating the function r = r(x) from expressions (91) and (98) as above we get the general one-dimensional singular integral equation with resepct to
1I
for Problem BL
40
J. M. Ra"ia.
which is:
lI(X)
111 (1 + 1- 2t ) lI(t)dt t - x t + x - 2tx
+-
1r
+
11
0
(108)
K(x, t)lI(t)dt = F(x) ,
where
K(x,t)
1
a
:=--a [G o (x;t,0)-ln(t+x-2tx)]. 1r x
(109)
Note: For a < x, t < 1 K(x, y) : continuously differentiable but at the endpoints of these intervals it may become infinite. In particular, however, whenever 91 terminates 2n short portions 00' and AA' of the semi-circle 9r, function K(x, t) will have no singulan't1'es at the endpoints of the said intervals. In this case the same can be said about the behavior of the function F(x) as about the right side of equation (106). Remarks: With respect to the kernel K(x, t) (in Tricomi's) in equation (52) the same may be said as above.
Lecture Notes on Mixed Type Partial Differential Equationll
6.
41
The Gellerstedt Problem
In 1935 S. Gellerstedt (Doctoral Thesis, 1935; Jbuch Fortschritte Math. 61, 1259) generalized Problem T by replacing coefficient y of u""" in equation (*) (Tricomi's equation) by
sgn(y)lylm,
m >
a.
The Gellerstedt Problem or Problem G: Consists in finding a function u = u(x, y) which satisfies equation
sgn{y)lylm u"""
+ U yy = a
(The Gellerstedt equation)
(G)
in a mixed domain D which is simply connected and bounded by a Jordan (non-selfintersecting) "elliptic" arc g1 (for y > 0) with endpoints a = (0, 0) and A = (1, 0) and by the "real" characteristics g2 : x
2
m.±.a.
2
m.±.a.
+ m + 2 (-y) , = 1
g3 : x - m
+ 2 (-y) , = a
of (*) satisfying the characteristic equation
(110) such that these characteristics meet at a point P (for y < 0) and assuming prescribed continuous boundary values (**). Consider the normal curve of Gellerstedt with equation
g~ or equivalently
(
1)
x-"2
2
4
+ (m+2)2Y
m+2
1
=4
42
J. M Ra8.ia8
where Z
such that
g1
contains
g~
. 2 .!!!..±.2 = X+l--Y m+2
2
in its interior.
Note: The definition for regularity of solutions of Problem G is identical to that one for Problem T.
Lecture Note3 on Mized Type Partial Differential Equation3
7.
43
The Frankl Problem
In 1945 F. I. Frankl (Izv. Akad. Nauk SSSR ser. 9, 2, 1945, 121-143) established a .generalizaton of Problem T and Problem G for the Chaplygin equation
K(y)u xz >
K(y)
+ U yy
= 0 ,
whenever
(CH)
>
y
The Frankl Problem or Problem F: Consists in finding a function U = u(x, y) which satisfies equation (CH) above in a mixed domain D which is simply connected and bounded by a Jordan (non-selfintersecting) "elliptic" arc g1 (for y > 0) with endpoints 0 = (0,0) and A = (1,0)' by the "real" characteristic 92: x=
l
Y
J-K(t)dt+l,
y
of equation (C H) satisfying the characteristic equation
- (J-K(y)f (dy)2 and by the non-characteristic curve
9~
+ (dX)2
= 0
(111)
emanating from point 0 lying
inside the characteristic triangle OAP and intersecting the characteristic 92 at most once (9~ may coincide with the characteristic 93 of (CH) near point 0): 93 : x =
-l
Y
J-K(t)dt
and assuming prescribed continuous boundary values
{
u=¢(s)on91' u =
1jJ(x) on
9~ .
(F)
y
x
p
Fig
3
Note: On g~ we must have
or
dy dx
1
g~:---<-
.J-K
dy\ g. < I
or
g~ : dx
or
g~ (as
:
+ V- K dy
1'" dx+ l
Y
>0
J-K(t}dt > 0
g~ emanates from 0
= (0,0))
0)
Lecture Note, on Mized TlIPe Partial Differential Equations
or
g~
x+
l
Y
45
J-K(t)dt > 0
or equivalently
g~
: x>
-l
Y
(112)
J-K(t)dt
Remarks: I. The Frankl Problem may be called also the Generalized Tricomi Problem or Problem GT. II. F. I. Frankl (Izv. Akad. Nauk SSSR ser. 9, 2, 1945, 121143) initiated a new stage in the theory of equations of mixed type. He showed that the problem of a supersonic flow out of a plane-walled vessel (the velocity inside the vessel is subsonic) can be reduced to the Tricomi Problem for the Chaplygin equation (CH): K(O) = 0, K'(y) > O. Besides Frankl studying transonic flows established the uniqueness of the solution of Problem F (or Problem GT) in the case where
F(y) = 1 + 2 ( : , ) ' > 0,
y<0
(113)
(Frankl's condition).
Note: The positiveness of F is equivalent to the convexity of ( - K) - } . More Applications 1. In 1959 I. N. Vekua (Fizmatgiz, Moscow, 1959; English Transl., Pergamon Press, Oxford; Addison-Wesley, Reading, Mass.,
46
J M. RaBlia.
1962) 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. 2. In 1961 M. N. Kogan (Prikl. Mat. Mech. 25, 1961, 132-137; J. Appl. Math. Mech. 25, 1961, 180-188) observed that the equations of magneto hydrodynamic flows in the region of sonic and Alfve'n velocities are equations of mixed type. 3. In 1953 F. I. Frankl (Kirgiz. Gos. Univ. Trudy Fiz. - Mat. Fak. 1953, no. 2, 33-45) noticed that the equations of the flow of water in an open channel at a velocity exceeding the propagation velocity of surface waves are also of mixed type. 4. In 1951 F. I. Frankl (Leningrad Univ. Ser. Mat. Meh. As-
tronom. 6, 1951, no. 11, 3-7) pointed out two gas-dynamical applications of the Bitsadze-Lavrentjev Problem. S. In 1959 F. I. Frankl (Izv. Vyss. Ucebn. Zaved. matematika, 1959, no. 6, 13, 192-201) reduced the problem of flow inside a plane- parallel Laval nozzle of given shape (the direct problem of the theory of the Laval nozzle) to a new boundary value problem for the mixed type equation (114)
i.
with m = Besides he showed that it is no longer sufficient to specify u = u(x, y) on 91 and on one of the characteristics in order to ensure the existence and uniqueness of solutions of equations (114) with 0 < m < 1. Finally instead of the usual continuity condition on the parabolic curve OA:
uy (x, +0) = uy (x, -0)
(115)
one requires the Frankl's discontinuity condition:
(FD)
Lecture Note. on Mixed Tllpe Partial Differential EquatioRl
47
Fundamental Extremum Principles 1. In 1950 A. V. Bits~dze (Dokl. Akad. Nauk SSSR 70,1950,561564) stated the following extremum principle: A solution of the Problem BL vanishing on the characteristic 93 achieves neither a positive maximum nor a negative minimum on an open arc OA of the type degeneracy cuve. 2. In 1951 P. Germain and R. Bader (C. R. Acad. Sci. Paris, 232, 1951, 463-465) established the above principle for the Problem
T. 3. In 1952 K. I. Babenko (Doctoral dissertation, Steklov Inst. Math., Moscow, 1952) proved the said principle for the Tricomi Problem when the equation (*) (the Tricomi equation) is replaced by equation yu xx
+ U yy + o:(x, y)u x + P(x, y)u y + r(x, y)u = 0 , if: o:(x,O)
(116)
= P(x, 0) = 0 .
Note: The above extremum principle is of great importance First: because it immediately implies uniqueness of the solution, and Second: because it enables us to apply the alternating method of Schwarz to solution of the Tricomi Problem under quite general assumptions on g1 4. In 1953 S. Agmon, L. Nirenberg and M. H. Protter (Comm. Pure and Appl. Math. 6, 1953, 455-470) established an interesting maximum principle for the equation K(y)u xx
+ U yy + o:(x, y)u x + P(x, y)u y + r(x, y)u = 0, if: K(y)
~
<
o
whenever
~
y
<
O.
(117)
48
J M. Ra"ia8
Besides they assumed that a twice continuously differentiable solution u = u(x, y) of equation (117), nondecreasing as a function of y along one of the characteristics, is defined in the characteristic triangle OAP. Finally they supposed that the following conditions held in OAP:
8 (
8(
r-K) r-:u
y-K
f3r-K) + a + r-:u y-K
- 2r
+ 2~
[8 (v'- K) + a + f3v' - K] [8 (v'- K)
+a -
f3v'-K]
~
(119)
0,
where
Then they proved that: If the maximum of u = u(x, y) is positive, it is attained on the interval OA of the parabolic curve. Note:
The uniqueness theorem for the Tricomi Problem for equation (117) follows easily from this maximum principle. In particular, if a = = r = 0 the uniqueness for Problem T holds if
f3
1 -+2 2
(K)' >0 K'
for
y < O.
(the Agmon-Nirenberg-Protter's condition) .
(120)
Lecture Note. on Mized Type Partial DiJJerential Equation.
49
This condition is more restrictive (i.e. stronger) than Frankl's condition (i.e.: F(y) > a for y < 0). Finally the' positiveness in (120) is equivalent to the convexity of (-K)- t,
Fundamental Bitsadze's Result In 1956 A. V. Bitsadze (Dokl. Akad. Nauk S.S.S.R. 109, 1956, 1091-1094) studied the Frankl Problem in the case that: both characteristics are replaced by two non-characteristics lying inside the characteristic triangle OAP. Fundamental Smirnov's Results 1. In 1963 M. M. Smirnov (Sibirsk Mat. Z. 4, 1963, 1150-1161) investigated the boundary-value problem for Gellerstedt equation (G) with the quantity y
m
dy au dx au ----ds ax ds ay
prescribed on 91 and the values of the unknown function on a characteristic. 2. In 1951, 1957, 1959 M. M. Smirnov (Belorusk. Gos. Univ. Ucen Zap. 12, 1951, 3-9; Vestnik Leningrad Univ. 12, 1957, no. 1, 80-96, 209-210; Vestnik Leningrad Univ. 14, 1959, no. 1, 130-133) investigated the fourth- order equation
a4 u ax4
+ 2sgn(y)
a
4u ax2ay2
a4 u
+ ay4
= 0
(or: U:, + sgn(y) :;,)' u= 0) In
(S)
a domain D bounded by a smooth curve gl in the upper
half-plane with endpoints 0 = (0,0) and A characteristics 92 Y = x-I g3
Y = -x
= (1,0), and by the
50
J M. RaAAiaA
of equation (S). Then he proved the existence and uniqueness of solutions of (S) satisfying the boundary conditions
Uig,
= rP1 (s),
au an I au an I
g,
g2
au an ig,
= rP2(S) ,
=
tP1 (x) ,
O~x~~,
=
tP2 (x) ,
~~x~l.
[S]
He also proved uniqueness theorems for boundary-value problems in the case that, besides: Uig, = rPl (s), ~~ ig, = rP2(S), the unknown function u = u(x, y) is prescribed on both characteristics, or the case that the unknown function is prescribed on one of the characteristics and the normal derivative an au on the other.
More Results 1. In 1957 L. 1. Chibrikova (Gos. Univ. Ucen. Zap. 117, 1957, no. 9, 44-47) obtained a solution of the Tricomi problem for equation (BL) in explicit form by using the properties of simple automorphic functions, doing this without use of conformal mappings when 91 is half the boundary of one of the fundamental domains of an elementary or Fuchsian group of Mijbius transformations.
2. In 1945, F. 1. Frankl (Izv. Akad. Nauk S.S.S.R. Ser. Mat. 9, 1945, 121-143; English trans!. techno memos Nat. Adv. Comm. Aeronaut. no. 1155, 1947) considered the problem of sufficiently wide supersonic jet flow impinging on a wedge, when a zone of subsonic velocities is formed ahead of the wedge; he reduced this to the boundary-value problem for the Chaplygin equation (CH), with: u = 0 on part of 91 near 0 and on the characteristic 93, and on the other part of 91 a homogeneous equation
P(x, y)u x
+ Q(x, y)u y =
0 ,
(121)
Lecture Note. on Mized TlIPe Partial Differential Equatioru
51
where P = P(x, y), Q = Q(x, y) are given functions. This problem is homogeneous and therefore determines the solution up to a constant factor. Frankl proved a uniqueness theorem for this problem, assuming condition:
F(y) = 1 + 2
(%,)' >
0,
y<0 .
3. In 1962 B. V. Melentev (Dokl. Akad. Nauk S.S.S.R., 143, 1962, 38-41; Soviet Math. Dokl., 3, 1962,338-342) proved the uniqueness of the solution to the Frankl problem for equations (*) and (BL), and also for a more general problem, in which (121) is replaced by P(x, y)u x
+ Q(x, y)u y + R(x, y)u =
where P = p(x, y), Q = Q(x, y), R and on the assumption that [P cos(nx)
0 ,
(122)
= R(x, y) are given functions,
+ Q cos(ny)]
R < 0 .
Melentev proved an existence theorem for the Frankl Problem for equation (BL) in D, when the curve gl has a special form. 4. In 1964 B. V. Melentev (Dokl. Akad. Nauk S.S.S.R. 154, 1964, 1262-1265; Soviet Math. Dokl. 5, 1964, 270-273) also studied the Hilbert-Poincare Problem for the Tricomi equation. S. In 1954, 1958 V. G. Karmanov (Dokl. Akad. Nauk S.S.S.R. 95, 1954, 439-442; Izv. Akad. Nauk S.S.S.R. Ser. Mat. 22, 1958, 117-134) used finite differences to prove the existence of a solution to the Tricomi Problem for equation (BL) under very weak restrictions on gl. 6. In 1965 L. I. Kovalenko (Dokl. Akad. Nauk S.S.S.R. 162, 1965, 751-754 == Soviet Math. Dokl. 6, 1965, 747-751; Dokl. Akad.
Nauk S.S.S.R. 162, 1965, 988-991 == Soviet Math. Dokl. 6, 1965, 789-793) used finite differences to prove the uniqueness of solution of the Tricomi Problem for the equation
sgn(y) Iylm h(y)uu
+ U yy + O!(x, y)u",
+ (3(x, y)u y + r(x, y)u =
f{x, y) ,
(123)
h{y) > 0 ,
m> 0,
under additional restrictions on g1 in the neighborhoods of 0 and A and on the coefficients O!, (3 , rand K:
K(y)
sgn(y)lylmh{y) .
=
Reduction Formulas Consider the Chaplygin equation
K{y)u"""
+ U yy = 0 , K{O) = 0 , K'(y) > 0
and characteristics g2
:
x=
g3
:
x=
l -l
rp1. =
2
Introduce new variables ~ =
x,
Y
Y
31
Then equation (CH) becomes
where
J-K{t)dt+ 1,
0
J-K{t)dt .
Y
JK{t)dt,
t> 0
(CH)
Lecture Notu on Mized TlIPe Partial Differential Equation.
Set u = z exp ( then
where
-i 1~
P(t)dt) ,
53
8.
Quasi-Regularity of Solutions
Definition 3. A function u = u{x, y) is a quasi-regular solution of Problem T for Chaplygin equation (in the sense of Protter) if
1) u E C 2 {D) n C{b) , 2) the integrals
exist, 3) Green's theorem is applicable to the integrals
JIv
uLudxdy,
JIv
u" Ludxdy,
JIv
u y Ludxdy .
4) the boundary integrals which arise exist in the sense that: the limits taken over corresponding interior curves exist as these interior curves approach the boundary,
5) u satisfies Chaplygin equation (CH) in D, 6) u satisfies boundary conditions
= (8)
on gl
,p{x)
on g2
=
(Protter's conditions or Adjoint Tricomi's conditions)
(P)
Lecture Notes on Miud Type Partial Differential Equatioru
55
9. The a, b, c Energy Integral Method and New Uniqueness Results for Quasi-Regular Solutions of Mixed Type Boundary Value Problems In 1953 M. H. Protter (J. Rat. Mech. & Anal., 2, no. 1, 1953, 107-114) used the so-called a, b, c-method, based on an idea of Friedrichs in order to establish uniqueness results for quasi-regular solutions of Chaplygin equation (CH) in a domain D with boundary conditions (P). The idea is the following: Consider u = u(x, y) be a quasi-regular solution of (CH) defined in D. Besides consider the integral
where a, b, c: sufficiently smooth functions of (x, y). By virtue of (CH) this integral vanishes. The functions a, b, c: are chosen in in such a way that, after G transformation of the integral by Green's formula, one obtains a positive (or non-negative) definite expression which vanishes only if u == 0 in D. Application: Take
Lu == K(y)u xx
+ Uyy + r(x, y)u =
!(x, y)
(124)
MuLudxdy ,
(125)
and boundary conditions (P)' where
Take J = 2{Mu, LU)D : = 2
JL
56
J. M. Rallia8
where
Mu :=a(x,y)u+b(x,y)u",+c(x,y)uy in
D.
(126)
Consider identities
2bruu", = (bru 2 )",
(br)",u 2
,
2cruuy = (cru 2 )y - (cr)yu 2
,
2aKuu""" 2auuyy 2bKu x u""" 2bu x u yy 2cKuyu""" 2cuyuyy
-
= (2aKuu",)", - 2aKu~ - (a", Ku 2 )", + a"""Ku2 , = (2auu y)y - 2au~ - (a yu 2)y + ayy u 2 , = (bKu~)", - b",Ku~ , = (2bu:tuy)y - 2byuxuy - (bu~)x + bxu~ , = (2cKu x uy )x - (cKu:) y + (cK)yu~ - 2c:tKuxuy = (cu~)y - CyU~ .
,
Besides employ Green's theorem:
t or
P(x, y)dx + Q(x, y)dy
Ii ~~ J"J{ D
dxdy
BP dxdy By
!'D
_~ -
f!
(~~ ~~) dxdy
=
D
Qdy
'~!aD Qv,d.
-
}
faD Pdx . - faD PV2ds
where
v
= (V1' V2)
: the outer unit normal vector on B D ,
s : arc length ,
(127)
(128)
Lecture Notes on Mixed Type Partial Differential Equatioru
57
Remember the Rule: (129) Then employing above identities and applying Green's theorem into relation (125) we get: J
=
JIv
[2aru 2 - (br)",u 2
(cr)yu 2
-
-
2aKu~ + a:z::z:Ku2 -
+ aY!lu2 - b",Ku~ - 2b!lu",u y + b",u;
+ (cK)!lu~
2au;
- 2czKuxu!I
-C!lU;] dxdy
+
1 laD
[bru 2 V1
+ cru 2V2 + 2aK UU'" V1
- ax K u 2 V1
+ 2auu!I V2
+ bKu~V1 + 2bu",u!lV2 - bU;V1 +2cKuxu!lV1 - CKU~V2 + CU;V2] ds .
- a!lu2v2
(130) Therefore
J
=
JIv JIv [(
[2ar - (br)x - (cr)!I
+
+(-2a
-2aK - bxK
+ bx -
+
1 laD
[2au(Ku"Y1
+
1 laD
[(bV1 - cv2)Ku;
+ (-bV1 + CV2)U~]
Case 1:
12
- 2(b!l
+ cxK)uxu!I
Cy)u~] dxdy
1 ~V1 + CV2)r] u laD
+
+- a!l!l] u 2dxdy
+ (cK)!I)u~
+
= 11
+ Kaxx
2ds
+ UyV2)
- (Ka x V1
+ a!lV2)u 2] ds
+ 2(bv2 + cKvduxu!I
ds
+ J 1 + J2 + J3
.
Uniqueness (The Adjoint Tricomi Case):
(131)
58
J. M. Ra"ia,
Assume Ul ,U2 : two solutions of Adjoint Problem T for equation (124) and adjoint boundary conditons (P). Then take (132) Claim that U
= 0 in
D.
(133)
In fact
Lu == K(y)u xx
+ U yy + r(x, y)u = 0
,
(134)
and {
u
= 0 on
[P]
u = 0 on
Therefore it is enough to show that u
= 0 on
(135)
93'
Then by a standard maximum principle we get (133). From (125) and (134) we get
J
= O.
(136)
From (131) and (136) to prove (135) it is enough to show that all integrals II ,12 , J 1 , J 2 , J 3 are non-negative.
First:
The integrals II ,12 are non-negative (~ 0) if the following two conditions hold in D:
= -2aK - b:z;K + (cK)y ~ 0 , = - 2a + b:z; - cy ~ 0 , = AB - (by
+ C:z;K)2
~ 0 .
Lecture Note. on Mixed Tllpe Partial Differential Equation.
Second.
59
The Integrals J 1 , J 2 , J 3 , are non-negative (~ 0) if the following four conditions hold on an : = g1 U g2 U g3
(b + cv'-K)r ~ 0
where: v
= (VI' V2)
on g3
the outer unit normal vector on g3,
b - cy' - K ~ 0 (equivalently:
bVI -
on g3 , CV2 ~
0
on g3)
Justification Conditions (Cd and (C2 ) hold obviously. Condition (C3 ): From [P] and the fact that
(137) we get
But
Therefore condition (C3 ) holds.
60
J. M. Rallia8
y
o >v, v
v
Fig
4
Conditions (C4 ) and (Cs ): J3
=
J
Qds
+
gl Ug,
where
Q
J
Qds : = J~l)
+ J~2)
,
g3
= Q(u x ,uy ) := (bill - cIl2)Ku~
+ 2(b1l2 + CKlldUxUy + (-bill + CIl2)U~ is a quadratic form with respect to From [P] we get
du = 0 or
or
Ux
,uy .
on gl U g2 ,
(138)
Lecture Note. on Mixed Type Partial Differential Equatioru
61
or (139) where
N : = normalizing factor . Therefore from (138): J~1) and (139) we get:
Qlrll u g.
:
=
[(b1l1 - c1I2)KlI~
+ 2(b1l2 + CKlId1l1112
+ (-b1l1 + C1I2)1Ii] N 2 or
But KlI~
+ 1I~ > 0
on g1,
and KlI~
+ 1I~ = 0
on g2
(141)
because K > 0 on g1 and g2 is characteristic. Therefore from (140)-(141) we obtain
if (C4 ) holds. Therefore J~ 1) ~ 0 if (C 4 ) holds. Also from (138) we have J~2) ~ 0 if (142) But on g3 :
62
J. M Ra8lia.
because K II~
+ II~
= 0
on g3
(as characteristic).
Therefore (142) holds if and or if
or if
or if condition (Cs ) holds (as 111 < 0 on g3). Condition (C6 ): From [P] we get
But dulY3 = u",dx + u y dy = u'" (-1I2ds) = (-
U'"
V- K + U
y )
+ U y (111 ds) (as 112 = 111 V- K on g3)
111 ds
or ( K U'" 111 + U y 112 ) ds V-K
Therefore
J~l) =
J Y3
2aV-Kudu=
J g3
aV-Kd(u 2 ) .
Lecture Notes on Mized Type Partial Differential Equatioru
63
By integration by parts we get
J~I) But
-1
=aJ-Ku 2Ig3
g3
u2d(av'-K) .
a on the upper and lower limits of 93 (as u(O) = a ,and ul g2 = a from [P]) ,
u=
therefore
J~I)=-1
g3
u2d(av'-K).
(144)
Besides
dalg. = axdx + aydy = ax (-V2ds) + ay (v1ds) = ( - ax v' - K
+ ay)
VI
ds
(as V2 =
VI
v' - K
on 93 )
or
Therefore
Thus
JJ2) =
1 g3
(v'-Kda)
(145)
u2
From expressions (143)-(145) we get J2 =
Then J 2
~
-1
g3
[d (av'-K)
+ v'-Kda]
u2
.
a if d (av'-K)
+ v'-Kda ~ a
on 93 .
(146)
But
d(aJ-K) Ig3
=
(aJ-Kt dx+ (aJ-K)y dy
=
(ax J - K) dx + (a yJ - K + a2~ ) dy
=
J - K [ax dx + (a y+ ~~) dY]
=
V-K [ax (-J-K) + (a y + ~~/)] dy
(as dXlg3 =
-J-Kdy) ,
or
Besides
J-Kdal g3
=
J-K (axdx + aydy) J-K [ax (-J-K) + ay]dy
=
J-K [axJ-K - ay] (-dy).
=
(148)
or
J-Kdal g3
From (147)-(148) we get
d (aJ-K)+J-Kdalg3
=
J-K (2a",J-K - 2a y+ ~:~) (-dy) . (149)
But
Expressions (146) and (149)-(150) yield condition (Ca ). Therefore all the integrals: II ,12 , J 1 , J 2 , J 3 ,:~ 0 if conditions (Gi),i = 1,2, ... ,6 hold.
Lecture Note, on Mixed Type Partial Differential Equatiom
65
Finally we must choose: "nice functions" a
= a ( X, y)
'. b = b( x, y)
so that all these conditions hold.
c
= c ( X, y)
in D
If this occurs then uniqueness
follows immediately.
Remark: Choose
a: = canst.
Then (146) is equivalent to
ad ( J - K)
~
0
on 93 ,
or a (
or a
J - K)' dy ~ 0 -K' r-v:dy 2y-K
~
on 93 ,
0
on 93 ,
or to: a : = canst. : ~ 0
on 93 .
Well-Known Choices: 1. (Frankl's choice): 1 a =-2
In
D,
and
b = cJ-K ,
b= c = 0
4aK
c=--
K'
for y ~ 0
for y ~ 0 .
Assume Frankl's condition:
F(y)
= 1 + 2 (:,)'
> 0,
y< 0 .
66
J. M Rassias
2. (Protter's choice):
a = _e Px coshy) a
= _e Px
l
b= 0
l
b = cJ-K
c= 0
l
for
l
4aK
for
c=--
K'
l
y~O y~O
l
Assume Generalized Frankl's Condition:
F{y) ~ 2
KJ-K K' f3
l
1r
1= -2-
Ym
l
Ym : = max. of the ordinates of points on g1
Note: Both choices (i.e. Frankl's and Protter's) work with r
= o.
Case 2: (The Ordinary Tricomi Case) Assume Ul ,U2 : two solutions of Problem T for equation (124) and Tricomi boundary conditions
(T)
= ¢>(s)
on gl
= ,p{x)
on g3 .
Then based on above (Case 1) discussion
[Tl
u= 0
on g1
U
g3
(151)
Therefore it is enough to show that U
First:
= 0
on g2
(152)
{Cd and (C2 ) are same as those in Case 1. Therefore integrals II l 12
l
:
~
in D.
Lecture Note, on Mized Type Partial Differential Equatiom
Second:
67
The integrals J 1 , J 2 , J 3 are ~ 0 if the follo~ing conditions hold on an := g1 Ug 2 Ug 3 :
(C4 )'
the same as in the Case 1 ,
b + cJ - K ~ 0
ax J - K
+ ay +
on g2 ,
aK'
4K ~ 0
on g2 .
Justification Condition (C3 )': From [T] and the fact that
dxig. = J-Kdy
or - 1I2dsig.
= J-K1I 1 ds
or (153) we get
1
J =
1
[(b -
g.
cr-K) r] 1I1u2ds
,
But
Therefore condition (C3 )' holds. Conditions (C4 )' and (Cs )':
where Q: the same as in (138).
68
J. M Ra,,;a,
From [T] we get as above (in (139)):
where
N : = normalizing factor. Therefore from (154) : J~1) and (ISS) we get:
(similar to that of (140)) . But Kv~
+ v~
> 0 on 91,
and Kv~
+ v~ = 0 on 93
(157)
because K > 0 on 91 and 93 is characteristic. Therefore Qlg,Ug32:0 if(C4 )'holds. Besides from (154) we have J~2) 2: 0
if
(158)
because K v~
+ v~
= 0
on 92
(as characteristic) .
Lecture Note. on Mixed Type Partial Differential Equation.
Therefore (158) holds if
or if
or if
or if condition (C5 )' holds (as 111 > 0 on g2)' Condition (C6 )': From [Tl we get
But dulg,
= u",dx + uydy = U:J: (-1I2ds) + U y (1I1ds) = (u x V- K + uy) 111 ds ( as 112 = -111 V- K ( - K U:J:
dUlg, =
+ U yV-K)
(K U X 111
111 ds
. r-Y
= -
y-K
+ U y 112) ds r-Y
y-K
Therefore
J~I) =
-1
2aV-Kudu
=
g,
-1
aV-Kd (u 2 )
y,
By integration by parts we get
J~I)
= -aV-Ku 2 Ig,
+
1 g,
on g2 )
u2 d(aV-K)
69
70
J M. Rauia.
Then similar to the Case 1 we get:
J?l =
J
u2d(aV-K)
(160)
g.
Besides
dalg. = axdx + aydy = ax (-V2ds)
= (ax V- K + ay) VI ds
+ ay (vIds)
( as V2
= -VI V- K
on g2 )
or
Therefore
Thus
J~21
= -
J
(V-Kda) u 2
(161)
g.
From expressions (159)-(161) we get
J2 =
J
[d (aV-K)
+ V-Kda]
u2
g.
Then J 2
~
0 if
d (aV-K)
+ V-Kda ~ 0
on g2 .
(162)
But
d(aV-K) I g, = (av-Kt dx+ (aV-K)y dy = (ax
V- K) dx + (a yV- K + a 2~ )
= V-K [axdx
+ (a y + ~~/) dy]
dy
Lecture Note. on MIXed Type Partial Differential Equatiom
71
or
d(aY-K) i go =FK[a x (Y-K) + (a y + ( as dxl g, =
V- K dy)
~~)] dy
(163)
.
Besides
+ aydy) (V -K) + ay] dy
V-Kdal g, = V-K (axdx
V-K [ax
=
.
(164)
From (163)-(164) we get
d (aV-K)
+ V-Kdalg,
V-K
=
(2a xV -K
+ 2ay + ~~)
dy (165)
But
dylg, =
IIl
(as illig, > 0)
ds lg , > 0
Expressions (162) and (165)-(166) yield condition (C6 )'.
Special Case: Choose
a: = const.
Then (162) is equivalent to
ad ( Vor a ( V-
or a
K)
~
0
K)' dy ~ 0
-K'
r-;;dy
2y-K
~
0
on 92 ,
on g2 ,
on g2
or to: a : = const. : ~ 0
on 92 .
(166)
72
J M Ra16ia,
Remark: In both cases if a : = constant in
D
then
a::; 0 in
D.
Well-Known Choices: 1. (Frankl's new choice): 1.In D a = --
2 ' and b = -c../-K ,
b = c = 0 for Y :? 0 , 4aK c = ~ for Y ::; 0 .
Assume Frankl's condition Y<
F > 0, 2. (Protter 's new choice): a = _e{3:Z cos (j'Y) , b = 0 ,
a = -e{3:Zx,
b = -c../-K ,
o.
for Y :? 0 4aK c = -for Y < 0 K' -
c= 0
Assume Generalized Frankl's condition
f3 " :>
0 as above.
Note: Also here both choices work with r=
o.
Lecture Note! on Mized Type Partial Differential Equatio1U
73
On the Exterior Tricomi and Frankl Problem F. G. Tricomi (1923-)' S. Gellerstedt (1935-)' F. I. Frankl (1945-), A. V. Bitsadze and M. A. Lavrentiev (1950-)' M. H. Protter (1953-) and most of the recent workers in the field of mixed type boundary value problems have considered only one parabolic line of degeneracy. The problem with more than one parabolic line of degeneracy becomes more complicated. The above researchers and many others have restricted their attention to the Chaplygin equation: K(y)ux.z + U yy = f(x, y) and not considered the "generalized Chaplygin equation": Lu == K(y)u xx + U yy + r(x, y)u = f(x, y) because of the difficulties that arise when r : = non - trivial (4 0). Also it is unusual for anyone to study such problems in a doubly connected region. In this paper I consider a case of this type with two parabolic lines of degeneracy, r : = non - trivial (4 0), in a doubly connected region, and such that boundary conditions are prescribed only on the "exterior boundary" of the mixed domain, and I obtain uniqueness results for quasi-regular solutions of the characteristic and non-characteristic Problem by applying the b, c-energy integral method in the mixed domain. The Exterior Tricomi Problem Consider
Lu == K(y)u xx
K E C 2 (-),
+ U yy + r(x, y)u = r Eel ( .),
f
f(x, y) ,
E CO (-) ,
(+)
and such that
K
= K{y) > 0 for y < 0 and y> = 0 for y = 0 and y = 1 , and <0
1,
for 0 < y < 1 .
Consider a mixed domain D which
doubly connected, contains the two parabolic arcs. Al BI , A2 B2 , with end points: Al = IS
J. M. Ra"ia.
7.(
(-1,1), B1
= (1,1), A2 = (-1, 0), B2 = (1, 0), and has boundary aD
= Ext(D) U Int(D)
,
Ext(D): exterior boundary of D : = ro U r~ U r 2 U r~ U .6. 1 U .6.~ , and Int(D): interior boundary of D : = r 1 U r~ U .6. 2 U D.~ ,with boundary curves:
ro:
"elliptic are" for y > 1 connecting points: Al , B1 , r~: "elliptic arc" for y < a connecting points: A2 , B2 , r 1: characteristic for a < y < 1, a < x < 1 emanating from point: 0 1 = (0,1) : dx = J-Kdy, or r 1 : x = J-K(t)dt , r~: characteristic for a < y < 1 , a < x < 1 emanating from point: O2 = (0,0) :
Io'"
1:
1:
Io'" dx = I; J-Kdy, r 2:
characteristic for a point: B1 (1,1) : dx = 1Y J-K dy, or r 2 : x = J-K(t)dt + 1 , characteristic for a < y < 1 , a < x < 1 emanating from point: B2 = (1,0) : dx = J -K dy, or r~: x = J -K(t)dt + 1 , characteristic for a < y < 1 , -1 < x < a emanating from point: Al = (-1,1): I~l dx = J-Kdy, or D. 1 : x == - 1Y J-K(t)dt - 1 , characteristic for a < y < 1 , -1 < x < a emanating from point: A2 = (-1,0) :
II'"
r~:
II'"
.6. 1
:
.6.~:
I;
or r~: x = J-K(t)dt , < y < 1, a < x < 1 emanating from
= I
I:
I;
I;
1:
I
I;
I;
:
I~l dx = J-Kdy, or D.~: x = J-K(t)dt - 1 , characteristic for a < y < 1 , -1 < x < a emanating from point: 0 1 = (0,1) :
.6.~:
dx = J-Kdy, or D. 2 : x = J-K(t)dt , characteristic for a < y < 1 , -1 < x < 0 emanating from point: O2 = (0, 0) :
D. 2
Io'"
1:
1:
Lecture Note, on Mind TWe Partial Differential Equationl
fa'" dx
= -
faY V-Kdy,
or
6.~:
x=
-
75
f: J-K(t)dt ,
Besides
where G 1 : upper elliptic region: = {(x, y) ED, Ixl < 1, y> 1} G~ : lower elliptic region: = {(x, y) ED, Ixl < 1, y < O} G 2 : right-hand side hyperbolic region: = {(x, y) ED, 0 < x < 1, 0 < y < 1} G~: left-hand side hyperbolic region: = {(x, y) ED, -1 < x < 0, 0 < y < 1} with boundary aG 11 : = r~ u (B2A2) U r~ u (B 1 0d u (02B2) ,
ro u (AIBd,
aG I aG 2
= =
aG~
= 6. 1 U 6.~ U 6. 2 U 6.~ U (0 1 Ad U (A 202)
r 1 U r~ U r 2
The above characteristic curves intersect at the following points:
r 1 n r~
= PI, r 2 n r~ = P2 for 0 < y < 1 and 0 < x < 1, and 6. 1 n 6.~ = P{, 6. 2 n 6.~ = P~ for 0 < y < 1 and -1 < x < O. Besides assume boundary conditions u = u = U
=
4>1 (S ) on r 0 , ,pI (x) on r 2 , tP3
(x)
(i.e. : u :
on
6. 1
(++)
,
= continuous prescribed values on Ext(D)).
The Exterior Tricomi Problem, or Problem (ET): Consists in finding a function u = u( x, y) which satisfies equation (+) and boundary conditions (++).
76
J. M. Rauia,
A New Uniqueness Theorem Assume the above-mentioned domain D C R2, and the conditions r~O
(Rd
on
Int(D)
xdy - (y - l)dx ~ 0
{
xdy - ydx
~
0
on
ra
r'a
on
" star-likedness"
+ xr", + (y - l)ry r + xr", ~ 0 2r
{
~
2r+xr", +yry
K' > 0
Gl
In
,
~ 0
0
In
Gl
In
G 2 UG~ G'1
In
and
K' < 0
In
G~ .
Then Problem (ET) has at most one quasi-regular solution in the mixed domain D. Proof.
We apply the b, c energy integral method (a = 0 in D)
and use (++). First, we assume equation
Ul ,U2:
two quasi-regular solutions satisfying
(+) and boundary conditions (++). Then claim that
It is clear now that
Lu == K(y)u xx
[+] [++]
+ U yy + rex, y)u =
u = 0
on
0 ,and
Ext(D).
It is enough to show that u =
Ul -
U2
= 0
on
Int(D).
Lecture Notes on Mixed Type Partial Differential Equatioru
y
x , -I
x,l
G'1
Fig
Second, investigate 0= J
=2
JIv
(bu x
5
+ cUy ) Ludxdy
where
(C)
{
b = x,
c=y-l
In
b = x,
c=O
In
G2
uG~
b = x,
c=y
In
G~
.
G1
Then consider the identities
= (bru 2 )x 2cruu y = (cru 2 )y 2bruu x
2bKu x u n ,
= (bKu;t
{br)x u2 , {cr)yu 2 , - bxKu; ,
,
77
78
J. M. Rauia.
2bu", Uyy = (2bu", Uy)y - (bu~) '" 2cKuyu u = (2cKu",u y),. -
+ b", U~ , (cKu~)y + (cK)yU~
,
2CUyUyy = (CU~)y - CyU~ . Then employing above identities and applying Green's theorem we obtain: 0= J =
Ii
[-(br)",u 2 - (cr)yu 2 +bxu~
+1
laD
+ (cK)yU~
[bru2vl
bxKu~
- CyU~] dxdy
+ cru 2V2 + bK U~ VI + 2bux UyV2
+2cKuxuyVl - CKU~V2
+ CU~V2]
-
bu~ VI
ds ,
where V=;.( VI , V2) . - (d Y , - dX) : outer unit normal vector on ds ds
aD.
Therefore 0= -
+
Ii Ii
[(br)x
+ (cr)y]u 2dxdy
[(-b",K+ (cK)y)u;
+1
laD [(bVI +cv2)r]u
+1
laD
[(bVI - CV2)
+ (b", -cy)u~]dxdy
2ds
Ku~ + 2 (bV2 + cKvd U",Uy
+ (-bVI + C/.I2) u~] ds II + 12 + J I + J 3 •
= Claim that all integrals: II ,12 , J I , and J 3 are non-negative.
First:
The integrals II ,12 are non-negative if the following two conditions hold in D:
Lecture Note, on Mixed Type Partial Differential EquatiofU
= -bxK + (cK)y ~ 0 ,
{~ Second:
19
= bx
-
Cy
~
0,
The Integrals J 1 , and J 3 are non-negative if the following conditions hold on aD: ~ 0
(bllr)r
:s:
bll l
0
on Int(D) ,
on Int(D) .
Justification
Condition (C3 )": From [++j and (C) we get
Therefore condition (C3 )" holds.
Condition (C4 )" and (Cs )": J3 =
r
QIds+j
iSxt(D)
Q 2ds
:=J~1)+J~2),
Int(D)
where = (bll l
Q2 : = Q2 (u x , uy)
+ (-bll 1 + Cll2) u~ , = (bllr) Ku; + 2 (bll2) uxuy + (-bllr) u~
-
clI2)Ku;
+ 2 (bll 2 + cKlIr}uxuy
Ql := Ql (u x , uy)
80
J M. Ra"ia6
are two quadratic forms with respect to
Ux , Uy
on Ext(D} , and
Int(D}, respectively. From
[++] we get Ux
=
NVl,
Uy
=
NV2
Ext(D) ,
on
where
N : = normalizing factor . Therefore
But
K
v; + v~ > 0
on
r 0 u r~
+ V~ = 0
on
Ext(D)\ro u r~
Kv;
(as
r2
(as K > 0
III
G1
U G~)
,r~ '~1 ,~~ are characteristics)
Therefore Ql =
QIIExt(D)
:= Qllrour~ ~ 0
if (C4 )" holds. Therefore J~ 1) ~ 0 if (C 4 )" holds. Also J~2) ~ 0 if
But on Int(D):
because
Kv;
+ v~ = 0
on
Int(D)
(as
r1
,r~ '~2 ,~~
are characteristics)
,
Lecture Notes on Mixed TWe Partial Differential Equations
81
From (C) therefore holds if ~
(blld K
a
- bll i ~
and
or if condition (Cs )" holds (as K < cation is complete.
a
Int(D) ,
on
a in G 2
U G~),
and the justifi-
Reduction of Conditions (Cd" - (Cs )" (by using choices (C)): Conditions (C3 )" and (Cs )" are reduced to condition: r ;:;
a
Int(D) ,
on
because XliI ;:;
a
on
I nt( D) .
Also condition (C.)" is reduced to condition: {
(R2) :
xdy - (y - l)dx ~ 0
on
ra
xdy - ydx ~ 0
on
r~.
,
Besides condition (C1 )" is reduced to condition: 2r + xr", + (y - l)ry ;:; { r+xrx;:;O 2r
+ xr x + yr y
a
0
;:;
In
G1
in
G 2 U G~
in
G~
Finally condition (C2 )" is reduced to condition:
K' >
a
in
G1
,
:
<
a
in
G~
because -K
+ (K + (y -
l)K') = (y - l)K' >
a
in G 1 -b",K
+ (cK)y
=
(if K' > -K > -K
a in
a in Gd
G2
+ (K + yK')
= yK' >
a
(if K' <
in G~
a
in G~)
82
J M. Rauias
and b", -
= 0 in G 1 U G~ , and bx
Cy
-
Cy
= 1 in G 2 •
Special Case:
(8) : K = sgn{y(y a,{3>O,
l»IYI'" Iy - liP h(y) h=h(y»O
and
in
forall
D, y,
where
sgn(y(y - 1) and: = 0 for y Therefore
K(y)
=
:= {
1,
y> 1
-1,
O
1,
y
= 0 and y = 1.
Kl(Y) = Y"'(y - l)Ph(y) > 0, { K2(y) = -y'" (1 - y)P h(y) < 0 , K3(y) = (-y)"'(l- y)Ph(y) > 0,
y> 1 0< Y < 1
y
and: = 0 for y = 0 and y = 1. Corollary If K = K(y) is of the form (8) in D1 if conditions (Rd-(R3) of Theorem hold1 and if
R = R{y j ex ,{3) = [a(y - 1) + {3y] h(y)
+ y{y -
l)h'(y)
in such that the following condition (B)
R >0
in
G1
,
R <0
m
G~
holds1 then Problem rET) has at most one quasi-regular solution in the mixed domain D C R2.
Lecture Notel on Mazed Type Partial Differential Equatioru
83
Remarks: 1). It is clear that on the parabolic lines of degeneracy: y = 1 and y =0: limy .... l+R(y;a,{3) = {3h(I) > 0, andlimy-+o-R(y,a,{3) = -ah(O) < a hold, because a, {3 > a and h(y) > a for all y in D. 2). If r : = constant, then conditions (Rd and (R3) are replaced by only condition (Rd. 3). If
a={3=I,
h=I
in (S), then
K(y) = sgn(y(y - I»lylly - 11 : = y(y - 1) and condition (B) in Corollary or condition (R.) in Theorem is not needed.
The Exterior Frankl Problem Replace characteristics r 2 , r~ , Al , A~ by smooth non-characteristics: g2 , g~ , 01 , o~ so that:
(NC) : emanating from point Bl lying inside the characteristic truncated triangle (0 1 , PI , P2 , Bl)' and intersecting r 1 at most once. This curve 92 may coincide with r 2 near point Bl ; ii). 9~ emanating from point B2 lying inside the characteristic truncated triangle (0 2 , B2 , P2 , PI) and intersecting r~ at most once. This curve 9~ may coincide with r~ near point B2 iii). 01 emanating from point Al lying inside the characteristic truncated triangle (Al , P: , P~ , ad, and intersecting A2 at most once. This curve 01 may coincide with Al near point Al ; iv). o~ emanating from point A2 lying inside the characteristic truncated triangle (A 2 , O2 ,P~ , P{) and intersecting A~ at most once. This curve o~ may coincide with A~ near point A 2 • i).
92
84
J M. Ra •• ia.
Besides assume boundary conditions
(F) :
1.t.
= 4>2 (s)
on
r~
u
= ~2(X)
on
9~
u = ~4(X)
on
8~
The new mixed domain DI is such that:
N ch (DI) = 92 U 9~ U 01 U
o~
: the
non-characteristic part of DI.
Besides
where
G2 (CG 2 )
:
={(x,Y)EDI ,
G~(CG~): ={(x,Y)EDI,
O<x
O
with boundary
r 1 U r~
9~ U (BlOd U (02B2) ,
8G 2
=
8G~
= 01 U o~ U ~2 U.6.~ U (OlAd U (A 2 02 )
U 92 U
•
The above non-characteristic curves intersect as follows:
The Exterior Frankl Problem, or Problem (EF): Consists in finding a function u = u(x, y) which satisfies equation (+) and boundary conditions (F) in the mixed domain DI.
Lecture Note. on Mized TlIPe Partial Differential Equatioru
85
Then it is clear that a corresponding new uniqueness theorem and a corollary hold in the new domain D' under the same conditions as those of the above proved theorem (and the corollary). The only difference in statement is that we must change C 2 UC; with C2 U C; in (R3)
(C)'
C2
(blld H
~
a
on
N ch(D')
In fact, from (NC): H > a on N ch(D'), the fact that b = x in· U C; (analogous to that one of (C)), and the fact that XliI>
a
on
N ch(D')
we obtain the validity of condition (C)'. This yields that we don't need to assume finally an additional condition in our new theorem (and corollary) in D'.
10. Weak(or Strong) Solutions in the Classical Sense Weak Solutions Let
q
1.
Ipi =
(p,q) = xx + yfj ,
= (x , fj)
{(p,p))2 ,
E ffi 2
0
Also let
for sufficiently smooth functions: u = u{p) Consider the generalized Chaplygin equation 0
Lu == K{y)u:u:
+ U yy + r{x, y)u =
K E C 2 {o),
r E
f
C 1 {o),
>
>
f{x, y) ,
K{y),
E C°(-)
(or
f
E L2 (o))
(EQ) g2 : may be either characteristic (Tricomi case):
~x dx= or
x=
l
Y
l
Y
J-K{t)dt,
J-K{t)dt
+1 ,
or a smooth non-characteristic (Frankl case) The domain G : = aD: = gl U g2 boundary of the mixed domain D C ffi 2
U g3 0
0
is a piecewise smooth
Lecture Notea on Mized TI/pe Partial Differential Equationa
87
Assume boundary conditions
(B):
u.=
a
on
g1 U g2
Also consider adjoint equation
(AQ):
L*w == K(y)w xx
+ Wyy + r(x, y)w =
f(x, y) ,
where L * (: = L) is the formal adjoint operator of the formal operatoI' L. y
x
p
Fig 6.
Assume now adjoint boundary condition
(AT):
w =
a
on
if
g2: characteristic, and
w =
(AF):
if
a
g1
on
U
G
g3
(Tricomi case)
(Frankl case)
g2: smooth non-characteristic.
88
J M. Rauia,
Denote
Cm(D) :
= {u(p)lpE D : u = u(p)
is m-times
continuously differentiable in Note: norm:
D} .
This space is a complete normed space with the following
II u Ilc"'(D) := max {I(D" u) (p)llp ED: lal
~ m}
Also denote
L2(D) := {u(p) Ip ED (U,W)L'(D) = and norm
Iv
with inner product
u(p)w(p)dp},
II u IlL'
(D)
=
(Iv lu(p)12
!.
dP)
2
Sobolev Spaces:
W;' (D)
= wm,2 (D) : = {u(p) Ip ED, u(p) E L2(D) , D"u(p) E L2(D) , lal ~ m} : is the Sobolev space with norm
II u 11m: =11 U Ilw mID) 2
such that
or also equivalently
II u 11m: =11 u Ilw;"(D) : = (II u
11~2
(D)
+
L
lal=m
II
Da
u
II~'(D)) t ,
Lecture Notes on Mixed Type Partial Differential Equation.!
and inner product
such that
(U,W)""
:= (U,W)w;,(O) :=
L
(Dau, DaW)L'(O) ,
lal:C:; ""
or also equivalently
(u, W)m
= (U,
+
W)w;, (0)
L
:=
(U, W)L'(O)
(Da u , DaW)L'(D) .
lal= ""
Remarks: 1. In general if 2
2
L
Lu -=-
Qi]
(P)Di D] u
+L
i,] = 1
Qi(p)Di U + Q(p)U
i= 1
then 2
2
L
L·w =
DiD] (Q;J(p)w) -
;=1
i,]= 1
11. Ill.
L D; (Q;(p)w) + Q(p)w
The above space W~ (D) is a complete normed space. W~(D)
c ... c WHD) c Wg(D)
11'lIm~
...
= L2(D),
~1I·1I1~1I·lIo
Special Case (: m = 2) : Let
D (L) : domain of the formal operator L :=
{uEC 2 (D): u=o on 91U92},
W;(D, bd) : closure of the function space D(L) with respect to the norm : = D(L)IIIII •.
II· 112
89
J M Ra .. ia6
90
see:
Ju. M. Berezanskii (Transl. Math. Mon., 17, AMS, Providence, R. I., 1968, p. 79-80). Similarly,
D(L*) : domain of the formal adjoint operator L*
:= {w E
C 2 (D) :
w= 0
on
gl U g3} (Tricomi case)
or
: = {w E
C 2 (D) :
w= 0
on
G} (Frankl case) ,
W~ (D, b* d) : = closure of the function space
with respect to norm
D (L *)
II· 112
:=D(L*)IIIII. '
or equivalently := {wlw E W~(D) ,(Lu,w)o = (u,L*w)o ,
for all u E W~(D, bd)} .
Definition 4. A function u E L2 (D) is a weak solution of Problem (EQ) & (B) if (J,w)o = (u,L*w)o
for all
w E W~(D,b·d).
Existence Criterion. A necessary and sufficient condition for the existence of a weak solution of Problem (EQ) and (B) is that (AP)
II
w 110::::: C
II
L*w 110
holds for all w E W~(D, b*d) ,
(a-priori estimate)
C = const. > 0 .
Lecture Note3 on Mized Tllpe Partial Differential EquationJI
see:
91
Ju. M. Berezanskii (Transl. Math., 17, AMS, Providence, R. I., 1968, p. 79-80; and J. M. Rassias ("MathematicsSpace Technology", Athens, Greece, 1981; "Partial Differential Equations of Mixed Type", manuscript at I.C.M.S.C. jS.P., Brasil, 1988, p. 12-27).
Definition 5. A function U E L2 (D) is a strong solution of Problem (EQ) & (B) if there is a sequence {un} : Un E C 2 (b) s~ch that
II Un
-
U
11- 0
and
II
LU n
-
f 11- 0 ,
n -
00
in the L2- norm in D. Remarks: i). {strong solution} c {weak solution}. i.e. a strong solution is a weak solution but a weak solution is not always a strong solution. ii). In 1958 K. O. Friedrichs (Comm. Pure Appl. Math., 11, 1958, 333-418) worked extensively on symmetric positive linear differential equations. iii). In 1960 P. D. Lax and R. S Phillips (Comm. Pure Appl. Math., 13, 1960, 427-455) proved that a weak solution is also a strong solution in the above classical sense or equivalently in the sense of Friedrichs (1958) by assuming local boundary conditons for dissipative symmetric linear differential operators. iv). In 1965 Ju. M. Berezanskii (Naukova Dumka, Kiev, 1965; Transl. Math. Mon., 17, AMS, Providence, R.I., 1968) developed a functional-analytic approach to existence proofs for weak solutions of the Tricomi and Frankl Problems. v). In 1966 N. G. Sorokina (Ukrain, Mat. Z., 18, 1966, 65-77) proved the uniqueness of the weak solution of the Tricomi Problem and showed that this solution coincided with the strong solution.
92
J. M Rauia.
vi). In 1980 J. M. Rasslas (Bull. Soc. Roy. Sci. Liege, 5-8, 1980, 278-280) established a new existence theorem for weak solutions of a mixed type boundary value problem with prescribed boundary values on a piece of the boundary of the hyperbolic region in the three- dimensional euclidean space. Uniqueness results for quasi-regular solutions of the above problem were established in 1977 via the doctoral dissertation of the same author (Doctoral Dissertation, U.C.-Berkeley, 1977). The generalization of these results in IRn+l(n ~ 2) was established by J. M. Rassias in 1988 (Comp. Rend. Acad. Bulg. Sci., 41, 1988,35-37; Compo Rend. Acad. Bulg. Sci., to appear). vii). We get an analogous Criterion for the existence of a weak solution if (AP) is replaced by the new a-priori estimate
II
[AP]:
W
111~ C
II
L*w
110 ,
because
II Note:
W
III ~II
w
110
In the !2-dimensional case:
II w Iii
=
Ji (w + w~ + w~) ~J i =11 II~ . 2
w 2 dxdy
dxdy
w
The Hahn-Banach Theorem and the Riesz Representation Theorem for Existence of Weak Solutions Note:
The following discussion is not necessary because the
Prove:
above-mentioned CrIterion is enough, but it was chosen as it gives a better understanding of the subject. If u E L2(D) and (AP)' or [AP] a-priori estimate holds, then u is a weak solution of Problem (EQ) & (B).
Lecture Notes on Mized Type Partial Differential Equatiom
Proof:
In fact
First:
Define the linear functional F in D (L *)
F : U = L * (D (L *))
F(L"w)
(Rd:
-+
93
IR ,
= (f,w)o
for all w E D(L"). Then
IF(L"w)1 = l(f,w)ol ~II
/
11011 w 110
(by Cauchy- Schwarz-
Buniakowski inequality) ~II
/
11011 w
III
(by inequality
II w 1I1~1I w 110
if [AP] a-priori estimate appears through, otherWise we apply (AP) a-priori estimate and don't consider this step at all; to go straight to the following step) ~II
/
110 C
II L*w 110
(by (AP) a-priori estimate) or
IF(L"w)1 ~ C
II /
11011 L"w 110
yielding that F: is bounded.
Note: U is a linear subspace of L2 (D). Second:
Employ the Hahn-Banach theorem to extend F from U onto the whole space L2 (D) with preservation of the norm.
In fact, there exists a linear functional
F : U = L2{D) --. IR as an extension of F (i.e. Flu = F) with preservation of Third:
the norm (i.e. II F 11&=11 F Ilu). Apply the Riesz representation theorem to find u E L2 (D) such that
F{u)=i uu and
II F 11=11 u IIV(D) for all
u E U.
Therefore
F(u) = (u, u)o
(by the definition of the inner prod uct (.) 0) .
Fourth:
Use the Hahn-Banach extension (above-mentioned). Thus,
F(u) = F(u)
for all
u E U( C
U) .
Therefore
F(u) =
Finally:
(u, u)o
for all
uE
U .
Choose
u=
L*w
for all
wE D(L*) .
Lecture Note, on Mixed Type Partial Differential Equationl
95
Hence
F(.L*w) = (L*w,u}o
for all
wE D(L*) .
Therefore relations (R)r -(Rh yield
(f, w}o = (L*w, u}o
for all
wE D(L*) ,
completing the proof that u is a weak solution of Problem
(EQ) & (B). Justification of the Definition of Weak Solution Assume u E C 2 (0) and
(f, w}o = (u, L·w}o for all wED ( L * ) . Claim that:
(i). Lu = f in (ii). u = 0 on
D gl Ug 2 .
In fact,
(f, w}o = (u, L*w}o for all w E D(L*), by assumption. First: By applying Green's theorem we get (Lu, w}o = (u, L* w}o +(u,w)c, or (f,w}o = (u,L*w}o = (Lu,w}o-(u,w)c (we do not know yet if Lu = f in D) for all wE D(L*) , where
(u, w)c
=
l
[w (KU%1I1 + Uy1l2)
-u
(kW:<1I1
+ WyIl2)] ds
,
where II = (111 ,112) is the outer normal unit vector on G.
Second:
Consider also the functional spaces: e~ (.0)(=
{wlw
E
em (.o)with compact support:
supp web} , supp
~------;-----'-----:--::"7
W
= {xlw(x) -I- a})
,
and
which is dense in
L 2 (D) (=> L· (D (L • )))
Third:
In particular, choose wE
e;;o (iJ)
(c D(L*))
such that w = \lw =
a
on
G
wE
e;;o (D)
and
(u, w)c
=a
for all
.
Therefore
(f, w}o
= (Lu, w}o
for all
w E e~ (D)
or
(f - Lu, w}o
=a
for all
wE
e;;o (D)
Lecture Notes on Mixed Type Partial Differential Equations
97
such that
w = Vw = 0
Fourth:
on
G.
Density of functions W(E C~ (b) in L2(b) implies
f
= Lu
L2 -sense .
in the
Continuity of Lu implies Lu =
Fifth:
f
in
D (that is validity of (i))
From (i) we get that:
(u,w)c=O
forall
wED(L*).
But since wE D(L*)
(i.e. w = 0
on G
Frankl case)
then
0= (u, w)c = = -
fa fa
U (KW"Yl
u (K lI;
+
+ Wyll2) ds
lin Nds
(as w = 0 on G yields II = I~:I : outer unit normal vector, or Vw = NlI , N : normalizing factor: = IVwl or w., = NlIl,
Wy
= NlI2)' or
J. M Rauia.
98
0=
(U,W)c =
-!
U
g,Ug,
(KII~ + lin Nds
(as 93 : characteristic: KII~ + 1I~ an arbitrary function. Therefore u = 0
on
=
0 on 93)' But N is
91 U 92 ,
completing the proof of (ii). Similarly the definition of weak solution is justified for the
Tricomi Case (i.e. w = 0 on 91 U 93)'
a-Priori Estimate We apply the a* , b* , c* energy integral method:
J* = 2{M*w,L*w}o
=
11
2M*wL*wdxdy,
where M*w : = a* (x, y)w
+ b* (x, y)w x + c* (x, y)wy
D,
III
and
Employing Green's theorem we get
where
I~
=
I;
=
J;
=
J;
=
11A~w2dxdy Ji (A;w~ t
,
- 2B*wxWy
t
B;w 2 ds ,
B; ds,
J;
=
t
+ A;w~) dxdy
Q* ds ,
,
Lecture Notu on Mixed Type Partial Differential Equatiol'U
A~
= (2a* - b; - c:) r - (b*r"
A;
= -2a* K - b:K
A*3
= -2a*
B"
= b;
B; B;
= =
+ b*"
+ c"ry) + Ka:" + a: y
+ (c* K)y
- c*y
+ c:K , (b*vl + C*V2) r , 2a*w (KW"Vl + WyV2)
- (Ka;vl
+ a;v2) w 2
and the quadratic form
Q* : = Q* (w", wy ) with respect to w" ,Wy :
Q" : = (b"Vl - c" V2) Kw~
+ 2 (b"v2 + c" Kvd W"Wy
+ (-b"111 + C"1I2) w~
.
Frankl Case: L*w = f,
First:
wiG = 0 .
On the boundary G:
J;
= 0,
J;
= 0 ,and
(as w" = N VI ,Wy = N V2 on G, N : normalizing factor) or
99
100
J M Ra.sia,
J;
=
1
+ C"V2) (Kv~ + v~)
(b" lI l
N 2 ds
01 UO,
(as 93 : characteristic: K
vt + v~ = 0
on 93) or
J*3 > - 0 if conditions
+ c" 112 ~ 0 on 91 K 1I~ + v; > 0 on 92
b" V1
U 92 , and (: non-characteristic)
hold, as Kv; + v~ > 0 on 91 (as K > 0 on 91)' or equivalently if conditions:
bOdy - c"dx
~
0
on
91
"star-likedness"
(R2F) : {
Second:
b" dy - c" dx ~ 0 0<
~ d
x
1
< v-K ,-
on 92 (: non-characteristic) on 92
hold. In the domain D: Denote
D2 = D n {y < O}
D1 = D n {y > O},
(OA) : = D n {y
= O}
.
Then
D
= D1
U
D2
U
(OA) .
Lecture Notes on Mixed Type Partial Differential Equation.
101
These Dl ,D2 are different from the Dl ,D2 (.= :" ,: = respectively). From
Note:
we get
But 2M*wL*w
= 2 (a*w + b*w" + c*wy) L*w = 2 (a*w) (L*w) + 2 (b'w,,) (L*w) + 2 (c*wy) (L*w)
Therefore
J*
Ii 21 ~ Ii ~ Ii {[Ill ~
M*wL*wldxdy
[2Ia*wIIL*wl
+ [1l3
(a*w)2
{C*Wy)2
+ 2lb*wx IIL*wl + 2lc*Wy IIL*wll dxdy
+
:1
{L*w)2]
+ [1l2
(b*W x )2
+
:2
{L*W)2]
+ 1113 (L*W)2]} dxdy
or
1 1 1) II + ( -+-+III
112
113
L*w II~
jill
,1l2 ,1l3 : > 0 .
But (as
J; = J; = 0,
and
J;
~ 0
from conditions (Rl F)-( ~ F)) ,
OOY'
102
J M. Ra88ia3
Therefore:
1;+1;
Ii [JLl(a*)2w2+JL2(b*)2W;+JL3(C*)2w~]dxdy
~
+ c~ II
L*w II~
where Cl
,
= ./~
VJLl
+ ~ + ~ := JL2
JL3
canst. : >
a.
Thus
(I; - Ii JLda*)2w2dXdy) + [I; - li[(JL2(b·)2w~ +JL3 (C*)2 W;)]dXd Y] ::;
C~
II
L*w II~
.
Therefore:
Ii
f3l w 2dxdy
~ c~ where:
II
+
Ii
(f32 W; - 2B*wxwy
L*w II~ f3l :=A~-JLl(a*)2, f32 :
= A;
- JL2(b*)2 ,
f33 : = A; - JL3(C*)2 .
Lemma. Denote
Assume
+ f33W~) dxdy
Lecture Notu on Mixed Type Partial Differentilll Equlltion8
where
kl : =k1(x,y) , = k2(X, y) ,
. k2
= k3(X,y)
k3
are given functions of x, y in D. Besides assume
2k : =kl
+ k3
kl
~
~ E:
>0
0
k3 ~ 0 in D. Then
Application: Take
k3
D
= {J3
= {JZ{J3 - (B*)2 .
Assume conditions in D:
{
(Jz ~ 0) {Jz
+ {J3
{J3 ~ 0 ~
E:
>0
103
104
J. M. Ra .. ia6
Then
Ii
f31w2dxdy+
Ii
(f32 W; - 2B*wxwy
~ c~
II w Iii,
C2 :
+f33W~) dxdy
= canst. : > 0 ,
Therefore
II will::;
[AP] : C
C
II
L*w 110 ,
= C:J../C'l.. . = canst.
>0 .
Remarks: 1.) If f32 = f33
= B* : = 0
Then
D = 0, and a-priori estimate [AP] (AP) :
II w
IS
replaced by a-priori estimate
110::; C
II
L*w 110 .
2.) If B* .= 0
then an a-priori estimate of the form [AP] holds immediately
(without employing above Lemma). In fact,
B* .= 0
Lecture
Note~
on Mized Tllpe Partial DiJJerenb.'al Equation.
if we choose: b*:=b*(x),
C*:=C*(y).
r:= canst.,
a*:= a*(y) ,
3.) If
then
Proof of Lemma:
where
>. : eigenvalues of matrix [M] of the quadratic form Q k Then
:= ( : :
.
or
. _ 2k ± J(2k)2 .
2
: = k ± .Jk2
- /)
4/)
:=k[l±Vl-~l But
105
106
J. M. Ra"ia.
Therefore
completing the proof of Lemma.
Remark: If [)=o
then
Amin : = 0 , and
In this case we have an a-priori estimate of the form (AP):
II w
110 ~ C
II
L* w 110
In fact, in this case
and
Qk : =
(yfk;w'" + ...jk;w
y
r:
where
as
Tricomi Case: L*w -
f ,
~0
(: = min. eigenvalue) ,
Lecture Nottl on Mixed Type Partial Differential Equation.
107
There are two main differences here from Frankl case: First: 92 is a characteristic. Second: w = a only on 91 U 93. The existence of a weak solution of Problem (EQ) & (B) can be found if we assume Conditions on Boundary G:
b* dy - c* dx
(R1T) :
~
a
on
91
"star-likedness"
(b* - c* r-K) r ~ a
1 b*
+ c*v-K
:::; 0
a: V- K + a; +
on 92 ,
a* K'
4K ~
a
on 92
and Conditions in D: Are the same as [R1 F]-[R3F] (in the Frankl case). Conditions for the Existence of Solution of the Boundary Value Problem
(EQ) :
Lu
==
[BJ :
K(y)uu
u
=a
+ Uyy + r(x, y)u = f(x, y) on
,
91 U 93 .
Then the adjoint boundary value Problem (Frankl case):
(AQ) :
L*w
==
K(y)w,u
+ Wyy + r{x, y)w =
f(x, y) ,
108
J. M Rauia.
[AF] :
w
=0
G.
on
In this case we assume conditions exactly the same as the conditions:
[Rl F] on gl, [R2 F] on g3 (: non-characteristic), and
Note: Here g2 is characteristic. On the other hand, the adjoint boundary value Problem (Tricomi case):
(AQ) :
the same as above, and
[AT] :
w
=0
gl U g2 .
on
In this case we assume conditions exactly the same as the conditions [RIF]-[R3F] in D, but on boundary G we assume here the following new conditions instead:
(Rl T) :
b* dy - c* dx
:2: 0 on gl (this is the same as (Rl T))
and "star-likedness"
(b*
(R2T) :
+ c*v'-K)
b* - c* v' - K
1
a: v' - K - a y
r
~
on g3 ,
0
:2: 0
on g3 , a* K'
+ 4 (-K )
~o
Note: Here both g2 , g3 : are characteristics.
on g3
Lecture Note. on Mazed Type Partial Differential Equation.
109
Application of the Energy Integral Method Separately in Dl and D 2 : Denote Dl = D n {y > O} ,
D2 = fj n {y < O} . Assume
a* E C 2(Dd n C 2(D2) , b* E C 1 (Dd n C 1 (D2) , c* E
C1(Dd n C 1(D2) .
Applying the energy integral method separately in Dl and D2 we get:
J;;l
Iii = Ii.
= 2(M*w, L*W)ODI :=
J;;. = 2(M*w, L*W)OD.
:
2M*wL'wdxdy, 2M*wL*wdxdy .
Employing Green's theorem in each case and then adding side by side we get:
J;;l + J;;. : =
jrJD,UD. r A~w2dxdy + j" r (A;w; - 2B*wxwy + A;w:) dxdy JD,uD. + J; + J; + J;
+
11
[((a:+ - a:_) - (c: - c:. )r)
w2
- 2(a: - a:' )wWy - 2(b: - b:' )WxWy -(c: - c:'
)w;] dx
.
Remarks: 1). Because of the last integral (: fol) we have to assume the following add£tional cond£t£on (to all the above cases concerning
11 0
J. M. Ra"ia.
uniqueness of quasi-regular solutions or existence of weak solutions):
(a: +
1
(KM) :
a: - ) - (c~ - c~) r ~ 0 ,
-
c~ - c~ ~
0 ,
a~ - a~ =
0 ,
b~ - b~
for all x : 0 where
~
x
~
= 0 ,
1,
( )+ = lim (
),
)_ = lim (
).
..... 0+ -+0-
y
x
Fig
7
2). On OA caDI: B;+
= {c: 1I2)r ,
Lecture Notu on Mized Type Partial Differential Equation,
Therefore
(i) :
l ' [(c~
rw 2) +
(2a~ wW II -
a;+ w 2)
OA
+(2b~ W",Wy :=
+ c~ w;)] l/2ds
1\),
Similarly On A 0 c B*1-
aD2 : = 2a_l/2WWII •
B*2-
all_l/2W • 2
= 2b:'l/2W"'W II + C:"l/2 W;
,
y
o
x
p
;/
Fig 8.
Therefore ii) :
r
lAo
[(c:' rw 2) + (2a:' wWII - a;_ w 2)
+(2b:' WXW II
+ C:' w~)J l/2ds
111
112
J M.
Rauia~
1I2ds = -dx
Adding (i) and (ii) replacing
11
[((a;+ - a;_) -
(c~ - c~)r) w
2 -
2(a~ - a~)wwy
-2(b~ - b~ )w",W y - (c~ - c~ )w;] dx .
3). Choose: a*
= {
y
y 2: 0
if
-y
if
y
~
:=
0
-Iyl
In
D.
and b* , c* b~
= b~
so that: , c~
= c~
Then we see that a~
but
= a~ : = 0,
. = {-I
ay
1
y 2: 0
if if
y
~
0
so that
a;+ - a;_
:=
-2(:< 0)
and condition (KM) fails to hold. But a* has to be chosen so that it is a C 2 (.) function. In our choice above a * is not C 2 (.). 4). The above additional condition (KM) is very important especially if the considered equation has discontinuous coefficients. In particular, in this case (with discontinuity) the energy integral method must be applied separately in D1 and D 2 •
Lecture Note8 on Mixed Type Partial Differential Equation8
113
A Uniqueness Theorem in a Three Dimensional Region In 1986 J. M. Rassias (Camp. Rend. Acad. Bulg. Sci., 39, 1986, 29-32) imposed t~e Bi-hyperboLic Bitsadze-Lavrentiev -Rassias equation
(*)
Ltl. = sgn{z) (tl.x:r; - tl.yy ) +
tl. ....
+ r{x, y, z)tl. = f{x, y, z) ,
and established uniqueness results for quasi-regular solutions. In particular, he considered the domain G in IR?, bounded by the s~rfaces:
S: : y + 1 = (x
2
+
y
1.
Z2)
2
,
= _{y2+Z2)t,I:: aG
that the boundary
aG Note
1.
Z2)
2
for z > 0,
4
x
and S:: x-I
= -(x 2 +
I: : y - 1 x+l
= {y2+Z2)t
for Z < 0, such
3
of G is given by = S: U
x
y
3
4
L U L U S:
y
x
4
3
s: n L = (AA'G), s: n L = (BB'G') and all the above surfaces intersect the {x, y}-plane at (ABA' B'). Besides, the surface So = (ABA' B') : = {(x, y) E IR? : Ixl + Iyl ~ I} is a parabolic degenerate surface for equation (*). Finally, G 1 : denotes that part of G above So (for z > 0) : = G n {z > O} and G 2 : denotes that part of G below So (for z < 0) : = G n {z < O}. Assume conditions
rx - ry ~ 0
III
G"
and
r =--
(I
In addition, assume boundat''i cond, wn u=O
on
J;
Y
3
4
LuI:
on
S,;
U
S!j
114
J. M Rauia.
z
y
Fig 9
Finally Rassias proved: Assume the above domain G c rn. 3 and conditions. Then Problem (*) and (**) has at most one quasi-regular solution u in G. Note:
That the case:
Lu = K(z)(u xx K(z)
>
-
U
yy ) + U zz
whenever
+ r(x , Y, z)u = z
>
f(x , Y, z)
2 (.)
I
I
was investigated through the doctoral dissertation of J. M. Rassias (U. C-BerkeWI 1977). Mixed Type Equations (Collection of Results) In the same year J. M. Rassias (BSB B. G. Teubner Verlags-
Lecture Note. on Mized Tllpe Partial Differential Equation.
115
gesellschaft, "Teubner-Texte zur Mathematik", L~ipzig, 90, 1986) collected most of the results on mixed type equations with applications in fluid dynamcis, This collection contains significant results by the expert researchers in the field of partial differential equations of mixed type: E. Ammicht & R. J. Weinacht (Newark), K. I. Babenko (Moscow), R. G. Barantsev (Leningrad), Chiu-Chun Chang (Taiwan), I. A. Chernov (Saratov), L. I. Chibrikova & N. B. Pleshchinskii (Kazan), G. C. Dong (Hangchow), M. Y. Chi (Wuhan), T. V. Gra~ chev (Sofia), Chaohao Gu & Jiaxing Hong (Shanghai), O. Jokhadze (Tbilisi), A. I. Kozhanov (Novosibirsk), M. Kracht (Dussseldorf), E. Kreyszig (Ottawa), A. G. Kuz'min (Leningrad), S: G. Mikhlin (Leningrad), A. M. Nakhushev (Nal'chik), S. Nocilla (Torino), I. E. Pleshchikskaya (Kazan), A. G. Podgaev (Novosibirsk), Ji Xinhua & Chen Dequan (Beijing), V. I. Zhegalov (Kazan), and the Editor (Athens). The contributions to this collection of works follow along the lines of the important work by F. G. Tricomi on boundary value problems of mixed type. Their originality and contact with many problems in fluid mechanics make this collection a most useful source of information about equations of mixed type. The topics covered include axially symmetric bodies, the Tricomi equation, the, Bitsadze-Lavrentiev equation, nonlinear problems, Galerkin's method, maximum principle, geometry, and gas dynamics. Through the above-mentioned collection of works J. M. Rassias (269-279) established uniqueness results for regular solutions.
The Chaplygin Equation In 1988 J. M. Rassias (Camp. Rend. Acad. Bulg. Sci., 41, 35-37, 1988) considered n
(E) :
Lu == K (xn+d
L i=l
U"'j"'j
+ u"'n+l"'n+l + r{x)u = j{x) ,
116
J M Rauia6
> < 0
K (Xn+d
Xn-t1
for
> < 0 ,
KI (xn+d > 0 .
K E C 2 ( .),
f
TEe 1 ( . ) ,
E CO (-)
2: 2 a simply connected multi-
Take as domain D c rn.n+ 1 , n
dimensional region bounded for x n + 1 > 0 by a smooth hypersurfa.ce 8 1 intersecting the hyperplane
11" : X n
+ 1 = 0 at
n
LX~
80
= 1,
i= 1
and for x n + 1 < 0 by two hypersurfaces 8 3 ,84 so that 8 3 is a smooth non-characteristic conic hypersurface intersecting the hyperplane
11"
at 8 0 with vertex on the x n + I-axis, and 8 4 is a character-
istic conic hypersurface intersecting 8 3 at 8~ (: with vertex at the origin 0, so that:
84
:
-=
(
LX~ n
)t + r+
;= 1
10"
Xn
+ 1 = t~+ 1 < 0)
1
J-K(s)ds=O,
0
where the "-" is used because 'V has to be outward on 8 4 ; v = (VI' V2,
..• , Vn , V n
ary G =
aD : =
+ 1) is an outer normal vector on the bound-
8 1 U 8 4 U 8 3 such that on 8 4 V=
It is clear that on 8 4 : i=1,2, ...
1 2
,n,
Lecture Note. on Mixed Type Partial Differential Equation.
.
..
.. '
. ,'.
5,
117
'.
.... . ' :: .... ... ' . ' . .. . ...' '... . . , ' " '......... . .. . . .. . . : . ' .. .... ',...... . . . '
"
~.
'
~o:~· ~. ~:;~:~:~.:- .-:-;:_ ~
. 0.
B = (1,0,0)
. .' '.:: ','
',.
I
I I
I I I
.
_-::"'_-=- -=-_~_!:.S;' -=-_-:.:-:. . P, '~:- - - - -:- - - -;-~ '2 -_-1 __ I
Fig
10
Graph for 3-dimensional Case (
n
= 2)
Assume boundary conditions (BC):
U
=
a
on
51 U 53 '
Mixed Type Problem or Problem (MF): Consists in finding a function U = u(x) which satifies equation (E) and boundary consitions (BC) in D. Uniqueness of Quasi-Regular Solutions Consider operator M: n
Mu == au
+L n=l
biu x •
+ cU"'n+1
In
D,
J. M. Ra66ia6
118
---
Graph for (n
Fig. 11.
+ 1)-Dimensional Case
(: n ~
where
(C) :
in D. Assume conditions: n
(Rl) :
L
,=1
X i V ,+CVn +1
~O on
SlUS3
2):
Lecture Note6 on Mixed Tllpe Partial Differential Equation6
119
Note: If 8 3 is characteristic then (Rd is assumed only on 8 1 -
a" -
(rc)x,,+~ + 2r (a -~)
n
- LXirXi ~ a
D,
In
i=l
a'
~
0
84
on
,
n
+ "'~+1
KL"'i2
> 0,
"'n+1
<
a
on
83
I
i=l
where (') means differentiation with respect to Denote in D:
Xn
+ 1-
n
A. = -2aK - (bdxi K + L(bj)xj K + (CK)X,,+l
I
i = 1,2, ___ ,n ,
j#-i n
,
- c , i=l
and on G(: = aD) :
n
Bn+1
= CVn+l
-
L i= 1
bi"'i,
Bij
= bjKvi + biKvj
,
J. M Ral$ia.
120
i i j :=1,2, ...
Bi =
CKl/i
+ bi l/ n + 1
,n,
i = 1,2, ...
,
Assume two quasi-regular solutions
Ul , U2
,n. exist for Problem
(MT). Then claim that =
U
U1 -
U2
= 0
D.
III
Therefore n
[E] :
Lu
== K(Xn+d
L
UX,Xi
+ UX,,+lX,,+l + ru = 0 ,
i= 1
and
[Be] :
u
=0
on
8 1 U 83
.
It is enough to show that u = 0
on
84
.
To prove this we apply Green's theorem in
0= J and get
0=
L(AoU' +
+
L[K t.
= 2(Mu, LU)D =
t.
i
MuLudx
.4,u;, + An", u;.+.) dx
2auu"v,
+ (2auu, •• ,
- a'u') vnH n
+2 = J1
il-J
t
+ J2 + J3
B'u" .... ) dS .
1dS
Lecture Notu on Mazed TlIPe Partial Differential Equation.
121
Then it is clear that all integrals J i , ; = 1,2,3 are non-negative, completing the proof for uniqueness of quasi-regular solutions.
Existence of Weak Solutions It remains to show the existence of a weak solution of Problem (M F). Assume additional conditions
do,e! := canst. > 0,
and
;=1,2, .. ,n, dj
R.
:= canst. > 0, Cz
a+ = Mo
j = 1,2,3,
= canst. > 0,
:
:
i = 1,2,3, ...
= canst.
n-2
+ --C2,
c+
2
>
,n,
a,
= d + K,
d:
= canst.
> a,
n
At
= 2 - {rc+)X,,+l
+ 2r (a+
- %cz) -
Cz
L rx,xi , i= 1
At
= 2K(K' - Mo)
+ dK',
A;
= 2{C2
Assume adjoint boundary condition w =
a
on
G
- Mo) - K'
III
D.
Besides assume condition n C2
~ Xi Vi
+ C+Vn +1
~ 0 on Sl
U
S3 .
i= 1
It is enough then to show that the following a-priori estimate holds IIwIl1~CIIL+wll, C :=const.>O, for all wED (L +) : = {w E C2 (D),
w = 0 on G}.
Note: II· II = II . 110, L + = L . To prove it he applied Green's theorem, Hahn-Banach theorem and Riesz representation theorem or a Criterion (necessary and sufficient conditions for existence of weak solutions). See: Ju. M. Berezanskii (Trans!. Math. Mon., A. M. S., 1968) and the corresponding 2-dimensional case in this book for further techniques and for the statement of the said Criterion. Then n
i= 1
where II =
Iv (.)dx ~
C1
II w II~,
C1
:
= canst. > 0 ,
for all w E D{L+), and
12
= fa OdS ~
0 .
Thus the a-priori estimate holds and the proof for the existence of a weak solution of Problem (GM) is completed. Therefore Rassias proved: Assume above domain D and conditions. Then Problem (M F) is well-posed in the sense that: there is at most one quasi-regular solution and a weak solution exists. Note: That the uniqueness part was carried out at U. C. Berkeley (1977) through the doctoral dissertation of J. M. Rassias.
Lecture Notu on Mized TlIPe Partial Differential Equation.
11.
123
Well-posedness
The Extended Chaplygin Equation In the same year J. M. Rassias (Comp. Rend. Acad. Bulg. Sci., 41, 1988, 35-37) considered the extended Chaplygin equation n
where
x = (X1,X2' ... ,Xn ,Xn +1) , >
X
n +1
<0 ,
Then he assumed the simply connected multi-dimensional "bellshaped" region D E IR n + 1, bounded for Xn + 1 > 0 by a smooth hypersurface 8 1 intersecting the hyperplane:
x n + 1 = 0 at 8 01 ,
and for Xn + 1 < 0 by two hypersurfaces 8 3 ,84 , so that 8 3 is a smooth noncharacteristic conic hypersurface intersecting the hyperplane: x n + 1 = 0 at 8 02 with vertex on the x n + I-axis, and 8 4 is a truncated characteristic conic hypersurface intersecting 8 3 at 8~ with vertex at the positive Xn + 1 -axis (only the truncated part of 8 4 for Xn +1 < 0 is considered). The outer normal vector v = (VI, V2 , ... boundary G = aD = 8 1 U 8 4 U 8 3 is such that
,Vn , V n
+ 1) on the
Denote
Do
= D n {x
D2 = Dn{x
Xn +1
= O},
Xn+l::; O}.
D1
= D n {x
Xn +1 ~ O} ,
124
J. M. Rauia!
5 02 \
x
I
\
I
\ \ \
I
,
,
5'o
I I
5'o
x = (xl,
Fig
X 2 ' ••• ,
xn )
12
Take SOl : L:~=I x; = mi, S02: L:~=I x; = m~ ,(ml ,m2 := const. > a : m1 < m2), and truncated (for Xn + I < 0) characteristic
- -((),,/..
r-r
,'f',Xn+1
)- {~EL -
PI
,
Pl
, ... ,
~ Pn
,Xn+1
} , where
= cos () cos
... cos
Pn = sin
E ~,
() E [0,211"] ,
j = 1,2, ... ,n - 2,
1
M,
= M o{s,9,¢) = (-
and
Pi
= [mi + i
= 1,2,
l
xn
+
I
tK;{S)P;'), ,
Ki(S)/Mo(s,(),
... ,n ,Pi>
a.
Lecture Notu on Mized Type Partial Differential Equationa
12 5
Assume boundary condition
(**) The Extended Mixed Type Boundary Value Problem, or Problem
(EF): consists in finding a function u = u(x) which satisfies equation (*) and boundary condition (**). Assume conditions
and
n
H =
L
KiV~
+ V!+l
< 0 on
S3.
;= 1
Therefore he proved the umqueness of a quasi-regular solution in D : Ao = -r"',,+l' A. = K~,i = 1,2, ... ,n, and on G : Bo = rVn+l,Bi = -K,Vn+l, j = 1,2, ... ,n, B n + 1 = vn+l,B i = KiVi,i = 1,2, ... ,n. Assume Ul, U2 two quasi-regular solutions, then claim that U = Ul - U2 = 0 in D. To prove this fact he applied the energy integral method in D, where Mu is defined above and Lu = O. Therefore, Green's theorem implies of Problem (EM). In fact, denote Mu
0= 2(Mu, Lu)v =
= u"',,+l
t
In (A,U' + A.u:,) dx + In (BoU' + t,B,U: + Bn+,u!." +2
t
B'U.,U•• +,) dS = J, + J, .
126
J. M. Rauia.
It is clear that both integrals J i ,i = 1,2, are positive. This completes the proof for the said uniqueness of 1./. in D. It remains to show the existence of a weak solution of Problem (EM). Assume additional conditions
clr
At
+ c+r"',,+1
= -(rc+ )",,,+1
~ 0
~ 0,
D,
III
and
= (c+ Ki)"',,+l
At
>0 ,
i = 1,2, ... ,n ,
Assume adjoint boundary condition w
= 0
on
G.
It is enough then to show that: A sufficient condition for the existence of a weak solution
1./.
E L2(D) of Problem (EM)
a-pr1'ori estimate holds
IIwll,$CIW w ll" " w
for all w E D(L*) ,
112
II wll:= = II w II~=
where
i (w'+ ~W~,)dX' Iv
C:= const. > O.
In fact, M·w = C·W"',,+1 , and
2'S
w 2 dx
the
Lecture Note, on Mlud TWe Partial Differential Equation.
I,
=
l
(~BiW:, +B~+, w;... + 2
t. B+'W.,W••• ,)
127
dS ,
where
and
.. ,
B +i=C+K-v-
.;-12 - , , ... , n .
It is clear that
11 ~ C 1 for all w E D(L+),
II W
II~,
(II w 11:5:11 wild ,
C 1 =: const. > 0
and
Thus the above a-priori estimate holds. Therefore the proof for the existence of a weak solution of Problem (EF) has been completed. Therefore Rassias proved: Assume above domain D and conditions. Then Problem (EF) is well-posed in the sense that: there is at most one quasi-regular solution and a weak solution exists.
The New Extended Chaplygin Equation In 1988 J. M. Rassias obtained a new result for well-posedness of a boundary value problem (in the sense that: there is a most one quasi-regular solution and a weak solution exists) for the new extended Chaplygin equation: n
(*)
Lu
=L i= 1
Ki(Xn+dux;x; +
uXn+1Xn+l
+ r(x)u = f(x) ,
12 8
J M. Rauia$
K 1(x n+d
>
<' 0,
Xn+1
>
<' a ,
for any Xn+1 E D (: = given domain),
K1(m) ~ -1,
where
m: = inf{xn+l
Xn +1 ED} .
Then he considered the simply connected domain D bounded for x n + 1 >
c IRn+ 1,
a by a smooth hypersurface 8 1 intersecting the
hyperplane: Xn +1 = Oat80l : y = (X2 ,X3, ... xn) = -(xg ,x~, ... , x~) = -yO,802 : y = yO,x~ := const. > O,j = 2,3, ... ,n, and two characteristic hypersurfaces
83
84
:
:
t/J = Xl
-
Xl
_l + x~ + l -
x~
of equation (107), x~ : = const. >
x
"+l
xn
1
+
a,
J-Kl(S)ds
= 0,
J-Kl(S)ds = 0,
both intersecting the negative
x n + I-axis at
Xn+l
= -X~+l
x~ +
l-
X~+l : = const. >
o Xn
+
1
J-KI(s)ds
= 0,
a.
The outer normal vector v =
(VI, V2,
..• ,V n , V n
+ I) on the
boundary G = aD = 8 1 U 8 3 U 8 4 U 8 01 U 8 02 is determined, as
Lecture Note! on Mixed Type Partial Differential Equation.
129
y
Fig
follows
Then he considered
13
130
J. M. Ra"ia,
' 1 '. Y -- _yO 80 8~ : 1/;'
=
Xl -
X~
' 8 02
:
Y = Y0 ,
+ Xn+1 = 0 ,
and 8 1 intersects the positive xn+1-axis at X n +1 = x~ Therefore the outer normal vector on 8 1 is: v
= V 1/;' /2VJ(; = ( 2yKl ~,0,0,
V
= Vt/>'/2VJ(; = (- 2yKl .~ v=
(0,- 2~'
v=
(0'2~'
~)
,0,0, ... ,0, . 2yKl
-2~' ... '-2~'0)
1
1)
2..JK;. , ... , 2.,fK;. ,0
Fig. 14.
Assume boundary condition
(**)
~)
... ,0, . 2yKl
on
on on
8~,
8~,
Lecture Note. on Mued TlIPe Partial Differential Equation.
131
The New Extended Mixed Type Boundary Value Problem or Problem (NT) consists in finding a function u = u(x) which satisfies equation (*) and' boundary condition (**). Assume conditions
r
< 0 on
S4, r
Note:
2r+ (Xl +x~)r"'l +Xn+l r"',,+l < 0
+ (Xl +
xn
r", 1
< 0
if
if
Xn+l ~,O ,
x+ I ~ 0 . n
That above conditions can be replaced by conditions
It is clear now by applying Green's theorem and conditions above that there exists at most one quasi-regular solution for the Problem (NT). To prove the existence of a weak solution of Problem (NT) assume conditions
if
Xn+l ~ 0 ,
where
A = X~
+ Ao,
Ao: = const. > 0,
J.I.:
= const. > 0
and
p(a+)2 ~ 0, Ao+ - .fLQ
A+ I
-
R I (b+)2 >0 I
,
,
132
J M Rauia.
where
R, : a+
= canst. > 0, 1
= - 2"'
c+ = Xn+l c+ = J.I.
if
bi
+ J.I.
i
= 0,1,2,
= Xl
In
+1 ,
D,
Xn+l ~ 0 ,
if
x n+ I
-).
... ,n, n
::;
0 ,
+ (rc+)x,,+J , - (bnXl Kl + (c+ Kdx,,+l , + (bnX1K J + (c+ KJ)X,,+l'
At = 2ra+ - [(rbi)Xl Ai = -2a+ Kl A;
=
-2a+ K j
A~+l = -2a+
+ (bnXl
- (C+)X,,+l
In
j
= 2,3,
... ,n ,
D,
and adjoint boundary condition
Then it is easy to show the following a-priori estimate
II w Ih::; C II
L+w 110,
C:= canst. > 0,
for all w E D(L+) : = {w E C 2 (D), w satisfies [**]} . Therefore the following result holds: Assume above domain D and conditions. Then Problem (NT) is well-posed in D in the sense that: there is at more quasi-regular solution and a weak solution exists:
Remarks: i. The Problem (NT) is the Tricomi Problem, or the Characteristic Problem. To investigate the Frankl Problem, or the Noncharacteristic Problem is not difficult and in this case someone must assume adjoint boundary condition on the whole boundary G.
Lecture Notes on Mixed Type Partial Differential Equations
133
ii. If someone considers an arbitrary hyper surface SI such that
on SI, in addition, then a new result holds.
The Extended Bitsadze-Lavrentjev Equation In the same year J. M. Rassias obtained another new result for well-posedness of a boundary value problem for the extended Bitsadze-Lavrentjev equation n+l
(*)
Lu = sgn(xn+1)uxlxl
+L
UXjxJ
+ r(x)u =
f(x) ,
3=2
where
sgn(Xn+d
=1
if
Xn+1 > 0 ,
=0
if
Xn+l=O,
= -1
if
X n +l
<0.
Then he considered the simply connected domain n c IRn+ 1, hounded for x n + 1 > 0 by a smooth hypersurface SI intersecting the hyperplane: x n + 1 = 0 at SO and for x n + 1 < 0 by two characteristic hypersurfaces S3 : tP = Xl - x~ + = 0,S4 : -tjJ =
(2:;:; x:)}
Xl
+ x~
-
(2:;:; x;)
12
of equation (*), x~ : = const.
intersecting the negative xn+1-axis at
,xn ). The outer normal vector v
Xn
+1
> 0, both
-x~. Denote: y =
(X2 ,X3, ...
IS: tJ
= (V1
,tJ2, ... ,vnVn+d on G
= an
J. M Rauia,
134
1
tJ
X2
= V4>/2 = ( - 2' 2(Xl + x~)
tJ
= V X/2
on
81
:
X3
, 2(Xl
X
+ x~)
Xn
, ... , 2(Xl
+ xn '
= X(x) = 0 .
Denote
Dl
= D n {x
:
Xn+l
> O},
D2
= D n {x
xn+1
< O} ,
y
x
Fig 15
Assume boundary condition
(**) and conditions
Lecture Note. on Mixed TIIPe Partial Diiferential Equation.
135
If someone applies Green's theorem in D. , i = 1,2 separately (: because of the discontinuous coefficient sgn(xn+d of U"'l"'.) and employs boundary condition (**) and above additional conditions he gets the uniqueness of a quasi-regular solution of the characteristic Extended Bitsadze-Lavrentjev Problem (BL): (*) and (**)" To prove the existence of a weak solut2"on of Problem (BL) assume the adjoint boundary condition
[**] and additional conditions
p(a+)2 Ao+ - .II{)
~
0,
A+1
-
R 1 (b+)2 >0 1
,
where
11. : =
canst. > 0,
i = 0,1,2, ... , n, n
1 a+ = - -
III
4'
c+ = Xn+l
At At
Ai A~+l
In
Dl
,
c+ = 0,
+1
D, In
Dz
+ (rc+ )", .. +1] , = sgn(Xn+d [-2a+ - (bi)"'1 + (c+)", .. +l] , = -2a+ + (bi)Xl + (C+)X .. +l' j = 2,3, ... = -2a+ + (bi) x l - (C+)X .. +1 In D.
= 2ra+ - [(rbi)"'1
,n ,
Then it is easy to prove the a-priori estimate
II will S; C II
L+ w
II,
C: = canst. > 0 ,
for all w E D(L +) = {w E C Z (D), w satisfies [**]} . Therefore we have the following result: Assume above domain D and conditions. Then the characteristic Problem (B L) is well-posed in D.
Lecture Note. on Mized TlIPe Partial Differential Equation.
12.
137
Open Problems
1. An open question concerns the regularity of solutions for the boundary value problems of mixed type discussed above. 2. In connection with Problem F of equation (CH) the main question remains of proving the existence and uniqueness of regular solutions without restrictions on K = K{y) or the size or shape of the domain. 3. No serious work is known on nonlinear boundary value proble!lls of mixed type in three and more dimensions. 4. The problem concerning the solution of elliptic systems in a domain on the boundary of which the type degenerates has not been investigated. 5. Little is known about the Cauchy problem for hyperbolic equations of order higher than 2 with boundary conditions on the curve of parabolic degeneracy. 6. The difficulty of the correct statement of the problem for equations of mixed type in higher dimensions still remains. 7. The study of higher order equations and systems of equations of mixed type requires more attention. 8. It would be interesting to clear up the question whether there is an extremal principle for the boundary value problems of mixed type. 9. One of the most important problems of mathematical physics is to study the properties of solutions of partial differential equations of mixed type with boundary conditions. 10. To consider regions of mixed type (elliptic-parabolic-hyperbolic): multi-connected with parabolic lines of degeneracy replaced by arbitrary curves (different from straight lines) is a question of high demand and difficult to handle. 11. The existence problem for regular transonic flow around given general profiles with given velocity at 00 is difficult to be solved completely. 12. Regarding the work of J. M. Rassias (Bull. Acad. Polonaise Sci. Ser. Ma.th., 28, 1980,311-313) there is still open question if n any odr; > 2.
138
J. M. Ra6.ia6
13.
Basic References
Basic Books 1. Ju. M. BEREZANSKII, "Expansion in eigenfunctions of selfadjoint operators", Amer. Math. Soc., Providence, R. 1. (1968). 2. A. V. BITSADZE, "Equations of the Mixed Type" , Acad. Sci. U.S.S.R., Moscow (1959); English Transl. Mac Millan Co., New York (1964). 3. J. M. RASSIAS, "Mixed Type Equations", BSB Teubner, Leipzig, 90, (1986). 4. , "Mathematics-Space Technology", Athens, Greece (1981). 5. , "Partial Differential Equations of Mixed Type", manuscript at LC.M.S.C./S.P., Brasil, (1988) and, "Boundary Value Problems of Mixed Type", manuscript at I.C.M.S.C./S.P., Brasil, (1988). 6. M. M. SMIRNOV, "Equations of Mixed Type", Moscow (1970); English Transl. Amer. Math. Soc., Providence, R.L (1978).
More Applied Books 1. R. G. BARAN CEV, "Lectures on transonic gas dynamics" , Izdat. Leningrad Univ., Leningrad (1965). 2. L. BERS, "Mathematical aspects of subsonic and transonic gas dynamics", Wiley, New York (1958). 3. C. FERRARI & F. G. TRICOMI, "Transonic Aerodynamics", Acad. Press (1968), English Transl. of "Aerodinamica Transonica" , Cremonese (1962). 4. K. G. GUDERLEY, "The Theory of Transonic Flow", Pergamon Press, Oxford (1962). 5. R. V. MISES, "Mathematical Theory of Compressible Fluid Flow" , Acad. Press, New York (1958). 6. F. G. TRICOMI, "Repertorium der Theorie der Differentialgleichungen", Springer-Verlag, Berlin (1968). Note:
Basic papers are mentioned through the topics.
SUbject Inde:z:
SUBJECT INDEX a-Priori Estimate 90., 98 Abel Integral Equation 16 Adiabatic Potential Flow 1 Adjoint Tricomi Case 57 Adjoint Tricomi's Conditions 54 Agmon-Nirenberg-Protter's Condition
48
Bitsadze-Lavrentjev Problem 33 Bitsadze-Lavrentiev-Rassias Equation
113
Cauchy Kernel 29 Cauchy Problem 9, 11, 15, 137 Cauchy-Goursat Problem 9, 13, 14, 30 Cauchy-Riemann Equations 4 Cauchy-Schwarz-Buniakowski Inequality 93 Chaplygin (Generalized) Equation 43, 45, 50, 73 Characteristic Coordinates 10, 11 Co-normal Derivative 18 Continuity Condition 46 Convexity 45 Energy Integral Method 55, 109 Euler Continuity Equation 5 Euler-Darboux Equation 11 Extended Chaplygin Equation 123 Extended Problem (BL) 135 Exterior Frankl Problem 83 Exterior Tricomi Problem 73 First Fundamental Functional Relation Formula of D'Alembert 35 Frankl Problem 43, 137 Frankl's Choice 65, 72 Frankl's Condition 65 Frankl's Discontinuity Condition 46
16, 35
139
14 0
Subject Indez
Fredholm Kernel 29 Fundamental Idea 9, 34 Galerkin's Method 115 Gas Dynamical Equation 1 Gellerstedt Problem 41 Generalized Frankl's Condition 66, 72 Generalized Tricomi Problem 45 Goursat Problem 10, 12 Green's Formula (Theorem) 17, 54, 56, 95, 109, 120 Green's Function 16, 36 Hahn-Banach Extension 94 Hahn-Banach Theorem 92 Holder Condition 13, 15, 23, 24 Hypergeometric Function 15 Integrals of Fractional Order Inversion Formual 15 Kernel
29, 40
Laval Nozzle 5 Liapunov Condition
36
Mach Number 4 Mobius Transformation
25
Normal Contour 17 Normal Curve 6, 33, 41 Open Problems 137 Ordinary Tricomi Case Problem BL 33 Problem (EF) 125 Problem F 43
66
15
Subject Indez
Problem G 41 Problem GT 45 Problem N 9, 16 Problem (NT) 131 Problem T 6 Problem of Complex Analysis Protter's Choice 72 Protter's Conditions 54 Quasi-regular Solution
26
54
Reduction Formulas 52 Regular Solution 8, 34, 42 Regularization of Singular Equation Riesz Representation Theorem 92
23
Second Fundamental Functional Relation Singular Integral 28, 29 Singular Integral Equation 22, 39 Sobolev Spaces 88 Strong Solution 91 Transonic Flow 5, 138 Tricomi Problem 6, 14 Tricomi's Theorem 23, 29 Well-posedness
123, 127, 132, 133, 136
18, 36, 37
141
Author 1ndes
AUTHOR INDEX Agmon, S. 47 Ammicht, E. 115 Babenko, K.I. 11,47, 115 Bader, R. 47 Barantsev (Barancev), R.G. 115, 138 Berezanskii, Ju.M. 90, 91, 122 Bers, L. 138 Bitsadz(!, A.V. 33,49, 73, 138 Carlemann, T. 24, 28 Chang Chiu-Chun 115 Chaplygin, S.A. 1 Chernov, I.A. 115 Chi, M. Y. 115 Chibrikova, L.I. 50, 115 Dequan Chen 115 Dong, G.C. 115 Ferrari, C. 138 Frankl, F.I. 43, 45, 46, 50, 73 Friedrichs, K.O. 91 Gellerstedt, S. 41, 73 Germain, P. 47 Gramchev, T.V. 115 Gu, Chaohao 115 Guderley, K.G. 138 Hardy, G.H. 13 Hong, Jiaxing 115 Jokhadze, O.
115
Karmanov, V.G.
51
143
144
Author Index
Kogan, M.N. 46 Kovalenko, L.1. 51 Kozhanov, A.1. 115 Kracht, M. 115 Kreyszig, E. 115 Kuz'min, A.G. 115 Lavrentjev (Lavrentiev), M.A. Lax, P.D. 91 Littlewood, J .E. 13
33, 73
Melentev, B.V. 51 Mikhlin, S.G. 115 Mises, V.R. 138 Nakhushev, A.M. 115 Nirenberg, L. 47 Nocilla, S. 115 Phillips, R.S. 91 Pleshchikskaya, I.E. 115 Pleshchinskii, N.B. 115 Podgaev, A.G. 115 Protter, M.H. 1,47,55, 73 Rassias, J.M. 1,91,92, 113, 114, 115, 122, 123, 127, 133, 137, 138 Smirnov, M.M. Sorokina, N.G. Tricomi, F.G. Vekua, I.N \"Veinacht r
49, 138 91
6, 29, 73, 138 45
T
, ~,
