International Journal of Applied Mathematics & Statistics,
Int. J. Appl. Math. Stat.; Vol. 13; No. J08; June 2008; 77-107
ISSN 0973-1377 (Print), ISSN 0973-7545 (Online)
Copyright © 2008 by IJAMAS, CESER
Mixed Type Partial Differential Equations
With Initial and Boundary Values in Fluid Mechanics
John Michael Rassias
Pedagogical Department E. E.,
Section of Mathematics and Informatics
National and Capodistrian University of Athens
4, Agamemnonos Str., Aghia Paraskevi
Attikis 15342, Athens, GREECE
E-mail: jrassias@primedu.uoa.gr, jrassias@tellas.gr
Abstract
This paper includes various parts of the theory of mixed type partial differential equations with
initial and boundary conditions in fluid mechanics ,such as: The classical dynamical equation of
mixed type due to Chaplygin (1904), regularity of solutions in the sense of Tricomi (1923) and in
brief his fundamental idea leading to singular integral equations, and the new mixed type
boundary value problems due to Gellerstedt (1935), Frankl (1945), Bitsadze and Lavrent’ev
(1950), and Protter (1950-2007 ). Besides this work contains the classical energy integral
method and quasi-regular solutions and weak solutions , as well as the well-posedness of the
Tricomi , Frankl, and Bitsadze - Lavrent’ev problems in the sense that: “There is at most one
quasi-regular solution and a weak solution exists”. Furthermore Rassias ( Ph.D. dissertation, U.
C. , Berkeley, (1977) ) generalized the Tricomi and Frankl problems in n dimensions based on
Protter’s proposal. This author generalizes even further the results obtained through the said
thesis. Also this paper provides a maximum principle for the Cauchy problem of hyperbolic
equations in multi-dimensional space-time regions, the formulation and solution of the TricomiProtter problem, a selection of several uniqueness and existence theorems and recent open
problems suggested by Rassias in the theory of mixed type partial differential equations and
systems with applications in fluid mechanics.
2000 Mathematics Subject Classification. 35K15, 76H05, 35M05
Key Words & Phrases. Chaplygin equation, Tricomi problem, Gellerstedt problem, Frankl problem,
Frankl condition, Bitsadze-Lavrent’ev problem, Rassias bi-hyperbolic
equation, exterior Rassias-Tricomi problem, general Tricomi-Rassias (or
GTR) problem, Tricomi-Protter problem.
1. INTRODUCTION
The theory of partial differential equations of mixed type with boundary conditions originated in the
fundamental research of Tricomi [63]. The Mixed type partial differential equations are encountered in
the theory of transonic flow and they give rise to special boundary value problems, called the Tricomi
and Frankl problems. The Transonic flows involve a transition from the subsonic to the supersonic
region through the sonic curve. Therefore, the transonic flows are very interesting phenomena
appearing in aerodynamics and hydrodynamics. The well-known mixed type partial differential
equation was called Tricomi equation : yu xx u yy
0 named after Tricomi ,who introduced this
equation, for functions u=u(x, y) in a real (x, y)-region. It plays a central role in the mathematical
analysis of the transonic flows, as it is of elliptic and hyperbolic type where the coefficient y of the
second partial derivative of the involved function u=u(x, y) with respect to x , changes sign. Besides
this equation is of parabolic type where y vanishes.
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International Journal of Applied Mathematics & Statistics
Definition 1. 1.The Tricomi problem or Problem T consists in finding a function u which satisfies
the afore-mentioned Tricomi equation in a mixed domain D : a simply connected and bounded (x, y)region by a rectifiable Jordan (non-self-intersecting) elliptic arc V (for y positive) with endpoints
O=(0,0) and A=(1,0) and by two real hyperbolic characteristics *, J of the Tricomi equation satisfying
the pertinent characteristic equation such that these characteristics *, J meet at a point P (for y
negative) with * emanating from A and
2
* : x ( y )3/ 2 1 and
3
2
3
J
from O,
J : x ( y )3/ 2
0
and u assumes prescribed continuous boundary values on both arcs
V
and
J
. The portion of D
lying in the upper half-plane, above the x-axis, is the elliptic region; portion of D lying in the lower
half-plane, below the x-axis, is the hyperbolic region; and the segment OA is parabolic.
Definition 1. 2. A function u=u(x, y) is a regular solution of Problem T if:
1) u is continuous in the closure of D which is the union of D with its boundary consisting of the three
curves V , *, J ;
2) The first order partial derivatives of u are continuous in the closure of D except
possibly at points O, A, where they may have poles of order less than 2/3;
3) The second order partial derivatives of u are continuous in D except possibly on OA where they
may not exist;
4) u satisfies Tricomi equation at all points of D except OA;
5) u assumes prescribed continuous boundary values on arcs
V
and
J
.
Fundamental idea of Tricomi in finding Regular Solutions for Problem T:
1) To solve Neumann problem or Problem N (in elliptic sub-region of D): To find a regular
solution of the Tricomi equation satisfying the boundary conditions:
1a) u assumes prescribed continuous boundary values on
function I on V ;
V
, such that, u equals to a continuous
1b) The first order partial derivative of u with respect to y equals to a continuous function Q
on OA except possibly at points O, A, where it may go to infinity of order less than 2/3.
Q ( x)
2) To solve Cauchy problem (in the hyperbolic sub-region of D):
To find a regular solution of the Tricomi equation satisfying the initial
conditions:
2a) u assumes continuous values on OA such that u equals to a continuous function
OA except possibly at points O, A, where it may go to infinity of order less than 2/3;
W W ( x)
2b) The first order partial derivative of u with respect to y equals to a continuous function Q
on OA except possibly at points O, A, where it may go to infinity of order less than 2/3.
on
Q ( x)
3) To solve Goursat problem (in the hyperbolic sub-region of D):
To find a regular solution of the Tricomi equation satisfying the boundary conditions:
3a) u assumes continuous values on OA such that u equals to a continuous function
OA except possibly at points O, A, where it may go to infinity of order less than 2/3;
3b) u assumes prescribed continuous boundary values on
function \ on J .
J
W W ( x)
on
such that u equals to a continuous
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
79
Most fundamental results on Mixed Type Equations with Applications:
In 1904, Chaplygin [2] has pointed out that the theory of gas flow is closely connected with the study
of a mixed type equation named Chaplygin equation
Lu : K ( y )u xx u yy
0
by replacing coefficient y of the second order partial derivative of u with respect to x in Tricomi
equation by a function K K ( y ) of y . Consider a two-dimensional adiabatic potential flow of a
perfect gas. The stream function \
\ ( x, y ) satisfies
( U 2D 2 \ y2 )\ xx 2\ x\ y\ xy ( U 2D 2 \ x2 )\ yy
where:
D:
the local velocity of sound and
U:
0,
(1.1)
the density of the gas.
This equation is transformed to a linear equation of mixed type by applying the following hodograph
transformation u
U 1\ y ; v U 1\ x ,where u, v : =the
rectangular velocity components as new
independent variables.
The corresponding components in polar coordinates are: r
u 2 v2 ; T
v
tan 1 ( ) .
u
(r / r0 ) 2 , which is dimensionless quantity, as independent variable,
where r0 : the speed corresponding to zero density. Therefore above-mentioned equation (1.1)
To normalize r , introduce : t
becomes
½ 1 (1 2 E )t
w ­ 2t
\t ¾
\ TT
®
E
E 1
wt ¯ (1 t )
¿ 2(1 t )
where:
E
0,
(1.2)
cv /(c p cv ) ,such that: c p :=the specific heat at a constant pressure,
cv : the specific heat at a constant volume. Note that:
r0
kJU01 /(J 1) : J
c p / cv , U0 : the density of gas at zero speed,
k (:=constant) satisfies the relation: p
kUJ .
Introduce new independent variables
[ T; K
Then, the afore-mentioned equation
³
t
1
2 E 1
(1 u ) E
du .
2u
(1.2) becomes
K (K )\ [[ \ KK
where: K
E:
0,
(1.3)
K (K ) (1 (1 2 E )t ).(1 t ) (1 2 E ) , and
a positive constant ( # 2.5 for air) ;
Besides: K (0)
0, because K
J:
a positive constant ( # 1.4 for air).
0 for t 1/(2 E 1) . This case corresponds to points where
the velocity is equal to the local velocity of sound, and therefore above equation (1.3) is parabolic.
K (K ) ! 0 , because K ! 0 for t 1/(2E 1) ;corresponding to subsonic velocities, and
(1.3) is elliptic.
Moreover:
equation
Similarly: K (K ) 0 , because
and equation (1.3) is hyperbolic.
K 0 for t ! 1/(2E 1) ;corresponding
to supersonic velocities,
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International Journal of Applied Mathematics & Statistics
Therefore ,the partial differential equation (1.3) is of mixed type.
Remarks 1. 3.
i.
The velocity potential
Riemann equations:
Ix
U 1\ y ; I y
I I ( x, y )
( U 2D 2 \ y2 )( U 2D 2 \ x2 ) (\ x\ y ) 2
D
\ ( x, y ) satisfy Cauchy-
U 1\ x .
The discriminant of equation: L\
ii.
and the stream function \
0 is given by the formula
( UD ) 4 (1 M 2 ) ,
M : Mach number : r / D ; r
\ x2 \ y2 / U .A flow is called subsonic, sonic or
supersonic at a point as the flow speed r D , D , or ! D .
where:
iii.
Transonic flows involve a transition from the subsonic to the supersonic region through
the sonic.
iv.
Equation (1.1) is quasi-linear and is converted to the linear equation of mixed type (1.3).
The corresponding equation is the following
quasi-linear equation: (D Ix )Ixx 2IxI yIxy (D I y )I yy
2
where
I I ( x, y )
2
2
2
0,
is the velocity potential. This equation comes from
the Euler continuity equation: ( UIx ) x ( UI y ) y
0.
In 1935, Gellerstedt [8] generalized Problem T by replacing coefficient y of the second order partial
derivative of u with respect to x in the Tricomi equation by a power of y. One may consider the wellknown
mixed
type
partial
differential
equation
called
Gellerstedt
equation:
m
sg n( y ) y u xx u yy
0, m ! 0
named after S. Gellerstedt ,who introduced this equation.
Definition 1. 4. The Gellerstedt problem or Problem G consists in finding a function u which
satisfies the afore-mentioned Gellerstedt equation in a mixed domain D : a simply connected and
bounded (x, y)-region by a rectifiable Jordan (non-self-intersecting) elliptic arc V (for y positive)
with endpoints O=(0,0) and A=(1,0) and by two real hyperbolic characteristics *, J of the Gellerstedt
equation satisfying the pertinent characteristic equation such that these characteristics *, J meet at a
point P (for y negative) with * emanating from A and J from O,
2
( y )( m 2) / 2 1 and
m2
*:x
J :x
2
( y )( m 2) / 2
m2
0
and u assumes prescribed continuous boundary values on both arcs
V
and
J
.
In 1945, Frankl [5] drew attention to the fact that the Tricomi problem is closely related to study of gas
flow with nearly sonic speeds.
Definition 1. 5. The Frankl problem or Problem F consists in finding a function u which satisfies
the afore-mentioned Chaplygin equation in a mixed domain D : a simply connected and bounded (x,
y)-region by a rectifiable Jordan (non-self-intersecting) elliptic arc V (for y positive) with endpoints
O=(0,0) and A=(1,0) and by one real hyperbolic characteristic * (from A),
*:x
³
y
0
K (t )dt 1, for y 0
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
81
of the Chaplygin equation satisfying the pertinent characteristic equation and by the non-characteristic
curve J c (emanating from O) lying inside the characteristic triangle OAP and intersecting the
characteristic
J : x ³
y
0
* at most once, where J c may coincide with the hyperbolic characteristic curve
K (t )dt , for y 0 , and * , J intersect at point P as well as
prescribed continuous boundary values on both arcs
V
and
u
assumes
Jc.
In 1950, Lavrent’ev and Bitsadze [14] initiated the work on boundary value problems for mixed type
equations with discontinuous coefficients. One can consider the well-known mixed type Bitsadze sgn( y )u xx u yy 0, named after Lavrent’ev and Bitsadze, who introduced
Lavrent’ev equation:
this equation with the discontinuous coefficient
sgn( y ) : 1, y ! 0; 1, y 0; 0, y
0.
Definition 1. 6. The Bitsadze - Lavrent’ev problem or Problem BL consists in finding a function u
which satisfies the afore-mentioned Bitsadze - Lavrent’ev equation in a mixed domain D : a simply
connected and bounded (x, y)-region by a rectifiable Jordan (non-self-intersecting) elliptic arc V
(for y positive) with endpoints O=(0,0) and A=(1,0) and by two real hyperbolic characteristics
*, J of :The Bitsadze - Lavrent’ev equation satisfying the pertinent characteristic equation such that
these characteristics *, J meet at a point P (for y negative) with * emanating from A and J from
O, * : x both arcs
J :x y 0
y 1 and
V and J .
and u assumes prescribed continuous boundary values on
The works performed in 1953, by Guderley [9] , Busemann [1], and Protter [16] are of great
importance in the field of mixed type equations with applications. In particular, Protter established
certain uniqueness theorems for boundary value problems involving equations of mixed elliptichyperbolic type by improving the famous Frankl condition
F ( y ) 1 2(
K ( y)
)c ! 0, for y 0 .
K c( y )
In 1954, Protter [17]showed how problems of Goursat type other than those considered by Soboleff
[62] and Garding [7] may be stated and solved for the multi-dimensional wave equation. These results
lead in a natural way to certain generalizations of the Tricomi problem for equations of mixed elliptichyperbolic type. Protter introduced the following definition:
Definition 1. 7. A function u=u(x, y) is quasi-regular solution of Problem T if:
D ‰ wD, wD V ‰ * ‰ J ;
u  C 2 ( D) ˆ C ( D), D
1)
1
2) The integrals
³ u ( x, 0)u
0
y
( x, 0)dx and
³³
D ˆ{ y ! o}
( Ku x2 u y2 )dxdy exist;
3) Green’s theorem is applicable to the integrals
³³
D
uLudxdy, ³³ u x Ludxdy, ³³ u y Ludxdy;
D
D
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 in D;
6) u satisfies prescribed continuous boundary conditions:
u I ( s ) on V ; and u \ ( x) on * .
2. UNIQUENESS AND EXISTENCE OF SOLUTIONS
In 1977, Rassias [20] generalized the Tricomi and Frankl problems in n dimensions, worked the
hyperbolic degenerate boundary value problem in n dimensions, as well, and introduced a new
boundary value problem in the theory of mixed type equations:
“The bi-hyperbolic degenerate problem” for the bi-hyperbolic degenerate equation:
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International Journal of Applied Mathematics & Statistics
K ( z )(u xx u yy ) u zz O ( x, y, z )u
Lu
0; 0 for z 0 and O d 0 ; Ox O y t 0 ,
K ( z ) ! 0 for z ! 0; 0 for z
in
f ( x, y, z ) such that:
{( x, y, z )} - bounded simply-connected region D( \ 3 ) surrounded:
S3y and a smooth surface 6 4y :
for z ! 0 by a characteristic surface
S3y : y x0
ª 2
«x ¬
³
1/ 2
0
ª
« y2 ¬
³
0
with non-negative real constants
All these surfaces intersect the
l1 : x y x0
1/ 2
2
º
K (t )dt »
¼
z
1/ 2
S 4x and a smooth surface 63x :
and for z 0 by a new characteristic surface
S 4x : x x0
2
z
ª
º
« x 2 U ³ K (t )dt »
0
¬
¼
2
º
K (t )dt » ; 6 4y : y x0
¼
z
; 63 : x x0
x
1/ 2
2
z
ª 2
º
« y U ³0 K (t )dt »
¬
¼
x0 ! 0; U t 1.
{( x, y )} -plane at the straight lines:
0 ; l2 : x y x0
0 ; l3 : x y x0
0 ; l4 : x y x0
0.
Then Rassias introduced the following mixed type boundary value problem:
3
The bi-hyperbolic degenerate boundary value problem in \ :A solution
u
u ( x, y, z ) of the above
63x ‰ 6 4y ,
bi-hyperbolic degenerate equation in D is found assuming prescribed boundary values on
such that
u
0 on 63x ‰ 6 4y and u  C 2 ( D) ; K  C 2 ( D) ; O  C1 ( D) ; f  C 0 ( D) ,
D
where
D ‰ wD ,and wD
S 4x ‰ S3y ‰ 63x ‰ 6 4y is the boundary of domain D.
Definition 2. 1. A function u u ( x, y, z ) is called quasi-regular solution of the above bi-hyperbolic
degenerate problem if the following conditions hold:
1)
u  C 2 ( D) ;
2) The integral
³ >.@dxdy
exists;
D ˆ{ z 0}
Di (H ) (i 1, 2) are regions with boundaries wDi (H ) lying entirely in Di such that
3) If
D1
D ˆ ^ z ! 0` ; D2
D ˆ ^ z 0` , the boundary integrals along wDi (H ) ,which result from
the Green’s theorem to the integrals
³³³ u dV ; ³³³H u dV
x
Di ( H )
y
(dV
dxdydz )
Di ( )
have a limit when
wDi (H ) approaches wDi .Then he stated and proved:
Uniqueness Theorem
2. 2. If we assume that
O d 0 ; Ox O y t 0
in mixed domain D, this bi-
hyperbolic degenerate problem has at most one quasi-regular solution in D.
In 1979 and 1982, Rassias ([21]-[27]) published the afore-mentioned part of his Ph. D. Dissertation
(U. C. Berkeley, 1977) on uniqueness results for the Frankl problem for his bi-hyperbolic degenerate
equation: Lu K ( z )(u xx u yy ) u zz O ( x, y, z )u f ( x, y, z )
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
83
and an application of positive symmetric systems [30].
In 1980, Rassias [22] introduced the bi-hyperbolic boundary value problem for his new bi-hyperbolic
equation: Lu K ( y )u xx sgn( y )u yy r ( x, y )u f ( x, y ) .
In
1982, Rassias [28] established maximum principles of the Cauchy problem for hyperbolic
3
equations in \ and
Sather
([59]-[60])
for
a
class
­
2
2
®( x, t ) : T0 d t d T 0, x1 x2 ¯
D
of
hyperbolic
f ( x, t )  C 0 ( D )
K (t )(u x1x1 u x2 x2 ) utt
Lu
in
\ n1 (n t 2) .His results generalized the results of Weinberger [64] and
³
T
0
,
equations
½
K ( s )ds ¾ ; D
¿
2
of
x
where
the
form:
( x1 , x2 )
;
D ‰ wD ,
{x1 , x2 , t} -space and analogously generalized in {x1 , x2 ,..., xn , t} - space.
Throughout this investigation, the influence of the work of Douglis [4] and of the famous book of
Protter and Weinberger [19] is apparent. In fact, let S0 be the portion of the boundary wD of D,
which lies in the plane
t T0 , and S4 be the remainder of the boundary G of D, which is a
characteristic conoid with respect to the above hyperbolic equation. Then Rassias introduced a
solution u u ( x, t ) of this equation in D which satisfies the initial (or Cauchy) conditions
the
change
of
variable
u ( x, T0 ) r ( x)  C 2 ( D), ut ( x, T0 ) h( x)  C 0 ( D). With
v( x, t ) u ( x, t ) r ( x) the afore-mentioned hyperbolic equation can be written as follows:
Lv K (t )(vx1x1 vx2 x2 ) vtt F ( x, t )  C 0 ( D ), where F F ( x, t ) f ( x, t ) K (t )(rx1x1 rx2 x2 ) .
The initial data above change to
v( x, 70 ) 0 vt ( x, T0 )
h( x)  C 0 ( D).
Assume the transformation to spherical polar coordinates:
x1
r cos p, x2
r sin p, where p  [0, 2S ), r
Therefore Rassias obtained
Lv
Let
r0
K (t )(vrr w2
w2
, 'v
wx12 wx22
'
1
1
v pp ) K (t )vr vtt
r2
r
F0
³
T
t
K ( s )ds.
vx1x1 vx2 x2
1
1
vrr vr 2 v pp ,
r
r
F0 (r , p, t )  C 0 ( D).
S 4e be the truncated part of S4 , where r0 ! e, such that
³
T
T0
K ( s )ds, e
³
T
T0
K ( s )ds : T0 d t d T 0 d T 0.
The direct characteristic conoid
characteristics of space-time: x1
S4 as well as the truncated one S 4e are generated by the bir cos p, x2 r sin p.
The angle parameter p is constant along each fixed generator of S 4 . On the other hand, the total
derivative of S 4 is given by
Setting d
d
dr
w
w
K (t ) , w
wt
wr
1
w
w
K (t ) .
wr
K (t ) wr
w(r , p, t )
r1/ 2 ( K (t )) 1/ 4 ; V
Rassias stated and proved the following maximum principle:
³
2S
p 0
v(r , p, t )dp on S4e ,
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International Journal of Applied Mathematics & Statistics
2
Maximum Principle 2. 3. Let us suppose that the function v ( r , p, t )  C ( D ) satisfies the differential
F0 d 0, and inequality : v(r , p, T0 ) 0, vt (r , p, T0 )
inequality Lv
h(r , p ) d 0,
where D is defined above, and assume the following condition:
d §
K c(t ) ·
d ¨w
¸ K (t ) d ( w) t 0, and : V
dr ¨© 2 K (t ) ¸¹
dr
In
Lu
0 . Then v(r , p, t ) d 0 in D.
1983, the author [31] introduced the bi-elliptic, hyperbolic and bi-parabolic equation:
y ( y 1)u xx u yy 0 with two parabolic lines: y 0 ; y 1 .
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 two parabolic lines of degeneracy becomes more
complicated. Rassias introduced an equation:
bi-elliptic for y
! 1, and y 0 ; hyperbolic for 0 y 1 ; and
bi-parabolic for
0 and y 1 .
y
Besides Rassias assumed D as a simply connected and bounded domain by:An elliptic curve
* 0 emanating from the points A1 (0,1) and B1 (1,1) and lying in the upper half-plane y ! 1 ; Another
elliptic curve *c0 emanating from the points A2 (0, 0) and B2 (1, 0) and lying in the lower half-plane
y 0 ; and
The following four characteristic hyperbolic curves of our above mixed type equation lying in the
region G2
{( x, y ) : x ! 0, 0 y 1}  \ 2 , such that:
*1 ( A1 P1 ) : x
³
*1c ( A2 P1 ) : x
³
y
1
1
S
;
(1 2 y ) y y 2 sin 1 (2 y 1) 4
8
16
y (1 y ) dy
1
y
y (1 y ) dy
0
y
* 2 ( B1 P2 ) : x 1 ³
1
*c2 ( B2 P2 ) : x 1 ³
y
0
1
1
3S
;
(2 y 1) y y 2 sin 1 (2 y 1) 4
8
16
y (1 y ) dy
1
1
S
;
(2 y 1) y y 2 sin 1 (2 y 1) 1 4
8
16
y (1 y ) dy
1
1
3S
.
(1 2 y ) y y 2 sin 1 (2 y 1) 1 4
8
16
If one considers the afore-mentioned mixed type equation, then Rassias introduces the Tricomi
problem which consists in finding a function u u ( x, y ) that satisfies this equation and the
boundary condition u |*0 ‰*0c ‰*2 ‰*c2
0 .Then he stated and proved:
Uniqueness Theorem 2. 4. If one assumed the star-likedness conditions:
­ xdy ( y 1)dx t 0 on * 0
,
®
¯ xdy ydx t 0 on *c0
then the Tricomi problem has at most one quasi-regular solution in D.
In
Lw
1988, Rassias [37] introduced the non-linear elliptic, hyperbolic and parabolic equation:
K ( y ) wxx wyy
and regular solutions w
f ( x, y, w, wx , wy ) and proved three uniqueness theorems for quasi-regular
w( x, y ) of the Tricomi problem.
In fact, he assumed the following conditions:
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
(c1 ) : K(0)=0;K c(y)>0 for y<0 ; lim
y o0
(c2 ) : ( p
K ( y)
K c( y )
85
0,
wy ) : are continuous functions of x, y, w, p, q : w, p, q  \1 ,
wx ; q
for any x, y in a simply-connected domain G bounded for y ! 0 by a non-self-intersecting Jordan
smooth curve *1 intersecting the
x axis (: y
0) at the points O1 (0, 0) ; O2 (1, 0) , and for y 0 by
the characteristic curves * 2 ; * 3 of the above non-linear equation emanating from the points
O2 (0, 0) ; O1 (1, 0) , respectively, and intersecting at some point in the lower half-plane, such that
*2 : x 1 ³
y
K (t )dt ; *3 : x
0
Denote G0
wG
G ˆ^ y
0` ; G1
³
y
0
K (t )dt .
G ˆ ^ y ! 0` ; G2
G ˆ ^ y 0` , and boundary
*1 ‰ * 2 ‰ *3 . The above-mentioned non-linear equation is elliptic in G1 ,
hyperbolic in G2 and parabolic in G0 . Also he assumed prescribed continuous boundary values:
g on *1 ‰ * 2 , where g g ( s0 ); s0  *1 ‰ * 2 , and g is continuous on *1 ‰ * 2 . The
author assumed two solutions w1 and w2 .
w
w1 w2 satisfies equation : Lu
Then u
Lu
r
where
s
s ( x, y )
³
t
t ( x, y )
³
h 1
h 0
h 1
h 0
'f
and
r ( x, y )
K ( y )u xx u yy 'f
0 , or
K ( y )u xx u yy r ( x, y )u s ( x, y )u x t ( x, y )u y
³
h 1
h 0
0,
(2.1)
f w ( x, y, w2 hu , w2 x hu x , w2 y hu y )dh ,
f p ( x, y, w2 hu , w2 x hu x , w2 y hu y )dh ,
f q ( x, y, w2 hu , w2 x hu x , w2 y hu y )dh ,
f ( x, y, w1 , w1x , w1 y ) f ( x, y, w2 , w2 x , w2 y )
f ( x, y, w2 u, w2 x u x , w2 y u y ) f ( x, y, w2 , w2 x , w2 y )
(because: u
'f
³
h 1
h 0
w1 w2 , or w1
w2 u ), or
df ( x, y, w2 hu , w2 x hu x , w2 y hu y )
u
ru su x tu y . Besides
0 on *1 ‰ * 2 .
(2.2)
Mixed type boundary value problem or Problem (M):
u ( x, y ) which satisfies equation (2.1) in a mixed type domain G
S
1
S 1
and assumes boundary conditions (2.2). Denote: E
, J
:
2 ym K ( ym )
2 ym
consists in finding a function u
where ym : maximum of the ordinates of points on *1 ;
f0
2
2
2
2
1 f p Kf q 2 K ( K E J )
,
4
K
KE 2 J 2
(
K ( y)
1)J 2 0 , as K is:
K ( ym )
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International Journal of Applied Mathematics & Statistics
y ym implies K ( y ) K ( ym ) ; R
K K
§ K ·c
R( y) 1 2 ¨ ¸ 2
E in G2 .
Kc
© Kc ¹
Assume the following additional conditions:
(c3 ) :
for ( x, y, w, p, q)  G1 u \3 ,
f w t f0
(c4 ) : R t 0 in G2 ,
(c5 ) :
K fq
fp
(c6 ) : 0 d f p d
for ( x, y, w, p, q)  G2 u \3 ,
1 Kc
R
4 K
for ( x, y, w, p, q )  G2 u \3 , and
3
(c7 ) : 4( K )( K c) 2 f w R t (4( K ) 2 f w K c f p ) 2 , f w t 0 , or
f 01 d f w d f 02
where
f 02
f 01
for ( x, y, w, p, q)  G2 u \ 3 ,
1 Kc ª
K c R 2 K f p K c K c R 4 K f p R º , and
»¼
8 K 2 «¬
1 Kc ª
K c R 2 K f p K c K c R 4 K f p R º .
»¼
8 K 2 «¬
Special Case: f p
0 (therefore : f q
0) : (c3 )c :
1
f w t ( K E 2 J 2 ) in G1 u \1 ,
2
2
1 § Kc ·
(c7 )c : 0 d f w d ¨ ¸ R in G2 u \1 .
4© K ¹
Then Rassias stated and proved the following uniqueness theorem:
Uniqueness Theorem 2. 5. If one assumed
(ci ) : (i 1, 2,3, 4,5, 6, 7) , then the above Problem (M)
has at most one quasi-regular solution in domain G.
To prove the uniqueness of quasi-regular solutions and thus above theorem one may consider the
differential operator Mu a ( x, y )u b( x, y )u y c ( x, y )u y :
a
­ 1 Ex
°° 2 e cos(J y ), in G1
; b
®
1
°
e E x , in G2
°̄
2
­° 0
®
°̄c K
, in G1
, in G2
and Green’s theorem on the double integral in G:
­ 0
°
® 4aK
°¯ K c
; c
Mu , Lu
G
³³
G
, in G1
, in G2
,
MuLudxdy .
Besides to investigate regular solutions we consider above conditions in a new simply connected
domain Ge ( part of domain G) bounded in the neighborhood of O1 by a circular arc De ( y>0) with
center O1 and radius e, and near O2 by Se ( y ! 0) : Se is the line: x
near O2 ; Se consists of the two the lines: x
characteristics:
dx
K dy .
1 e ; y
e , if
1 e , if
x 1 on *1
x t 1 on *1 near O2 ,and by
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
Denote: G1e
Ge ˆ ^ y ! 0` , G2 e
Ge ˆ ^ y 0` , G0 e
Ge ˆ ^ y
87
0` , and boundaries:
* 2 ‰ *cc3e ‰ G0 e ‰ *c3e ;
wG1e
*1 ‰ De ‰ G0 e ‰ Se , wG2 e
wGe
*1 ‰ * 2 ‰ De ‰ *c3e ‰ *cc3e ‰ Se .
Assume the following regularity condition: For some constant c1 ,
(c8 ) : K c( y ) / K ( y ) d c1 ( y )D 1 on *3
u ( x, y ) of Problem (M) is called regular solution if it (1). satisfies
u
Definition 2. 6. A solution
near O1 ; D ! 0, c1 ! 0 .
equation (2.1) in G except G0 ;
(2). is continuous in G;
(3). has continuous first derivatives in G and *1 , * 2 ( both open ) with the possible exception of the
points O1 , O2 , in whose neighborhoods they may have poles of order less than
ux , u y
o Oi P
Zi
, 0 d Z 1;
i
1:
Oi P o 0 (i 1, 2), P  G, and
(4). has continuous second derivatives in Gi (i
1, 2) with the possible exception of points on the
parabolic curve, in whose neighborhoods they may not exist.
Uniqueness Theorem 2. 7. If one assumed conditions (ci ) : (i
1, 2,3, 4,5, 6, 7,8) , then the above
Problem (M) has at most one regular solution in domain G.
In
1990,
equation: Lu
Rassias
[39]
introduced
K ( y )u xx u yy r ( x, y )u
the
bi-elliptic,
bi-hyperbolic
and
f ( x, y ) with two parabolic lines: y
bi-parabolic
0 , y 1 .These
boundary value problems with boundary conditions on the exterior boundary of a mixed type doublyconnected domain generalize his above uniqueness results [31]. He noted that this doubly-connected
domain D G1 ‰ G1c ‰ G2 ‰ G2c consisted of the following four regions:
G1 ( R1 {( x, y ) : x 1, y ! 1}) :”upper elliptic region” ,
G1c( R1c {( x, y ) : x 1, y 0}) :”lower elliptic region” ,
G2 ( R2
{( x, y ) : 0 x 1, 0 y 1}) :”right hyperbolic region” ,
G2c ( R2c
{( x, y ) : 1 x 0, 0 y 1}) :”left hyperbolic region”,
with “boundaries”: wG1
wG2
* 0 ‰ A1 B1 , wG1c *c0 ‰ A2 B2 ,
*1 ‰ *1c ‰ * 2 ‰ *c2 ‰ O1 B1 ‰ O2 B2 , wG2c
“edge” points: 21
(0,1), 2 2
{( x, y ) : x 1, y
(1,1), $ 2
(1, 0), %1
0} :”lower parabolic segment”,
21%1 {( x, y ) : 0 x 1, y 1} :”right upper parabolic segment”,
2 2%2
(1,1), %2
{( x, y ) : x 1, y 1} :”upper parabolic segment”,
“parabolic” segments: $1%1
$ 2%2
(0, 0), $1
'1 ‰ '1c ‰ ' 2 ‰ 'c2 ‰ O1$1 ‰ O2 $ 2 ;
{( x, y ) : 0 x 1, y
0} :”right lower parabolic segment”,
(1, 0) ;
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International Journal of Applied Mathematics & Statistics
21$1 {( x, y ) : 1 x 0, y 1} :”left upper parabolic segment”,
22 $2
{( x, y ) : 1 x 0, y
0} :”left lower parabolic segment” ;
“elliptic star-liked ” arcs: * 0 :”connecting points $1 , %1 in G1 ”, *c0 :”connecting points $ 2 , % 2 in G1c ”;
and eight “hyperbolic characteristic” lines of the above bi-hyperbolic equation:
*1 ( 2151 ) : x
³
y
K (t )dt , *1c ( 2 2 51 ) : x
1
* 2 ( %15 2 ) : x 1 ³
y
*1 ˆ *1c and 5 2
'1 ( $151c ) : x
1 ³
' 2 ( 215c2 ) : x
³
where 51c
y
1
y
1
y
K (t )dt ;
0
³
K (t )dt , *c2 ( %2 5 2 ) : x
1
where 51
³
y
1
K (t )dt ,
* 2 ˆ *c2 , as well as:
1 ³
K (t )dt , '1c ( $ 2 51c ) : x
K (t )dt , 'c2 ( 2 2 5c2 ) : x
'1 ˆ '1c and 5c2
y
0
³
y
0
K (t )dt ;
K (t )dt ,
' 2 ˆ 'c2 . Then Rassias stated and proved:
Uniqueness Theorem 2. 8. If one assumed the star-likedness conditions:
­ xdy ( y 1)dx t 0 on * 0
,
®
¯ xdy ydx t 0 on *c0
and the coefficient conditions:
­2r xrx ( y 1)ry d 0 in G1
°
; r d 0 on Int ( D) ; K c ! 0 in G1 ; K c 0 in G1c ,
® r xrx d 0 in G2 ‰ G2c
° 2r xr yr d 0 in Gc
x
y
1
¯
Int ( D) *1 ‰ *1c ‰ ' 2 ‰ 'c2
(interior of D), as well as
boundary conditions:
u 0 on Ext ( D) ,where Ext ( D) * 0 ‰ *c0 ‰ * 2 ‰ *c2 ‰ '1 ‰ '1c (exterior of D), such that:
wD Ext ( D) ‰ Int ( D) (boundary of D), then this Tricomi problem would have at most one quasi-
where
regular solution in D.
Similarly Rassias investigated the pertinent Frankl problem. Besides he considered the following
interesting special case:
K
K ( y ) sgn( y ( y 1)) y
D
E
y 1 k ( y ) ( D ! 0, E ! 0 ; k
k ( y ) ! 0 ) in D.
Besides Rassias [33] (in 1985) investigated the pertinent Bitsadze-Lavrent’ev problem with two
parabolic lines of degeneracy and two elliptic arcs. In 1990, he [40] established the well-posedness of
Tricomi-Bitsadze-Lavrentjev problem for the “generalized Chaplygin equation” with discontinuous
coefficient: K ( y ) sgn( y ) .
In 1990, Rassias [41] suggested twelve open problems, most of which still remain unsolved, and
variations of which are stated at the end of this work.
In
1992
Lu
K ( y )u xx u yy r ( x, y )u
arc g1
,
Rassias
[43]
considered the
elliptic-hyperbolic and parabolic equation
f ( x, y ) ,in a simply-connected domain G bounded by an elliptic
( for y ! 0) : connecting points :
characteristic curves:
A (1, 0) ;
Ac (1, 0)
and
by two hyperbolic
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
g 2 ({ PA) : x 1 ³
y
K (t )dt ; g3 ({ AcP ) : x
0
intersecting at
P { P (0, y p ), where
1 ³
y
K (t )dt ( for y 0) ,
0
y p ( 0) :
2
³
yp
K (t ) dt
0
89
1 , proved
a uniqueness
theorem on quasi-regular solutions in C (G ) ˆ C (G ) and the pertinent existence theorem on weak
2
solutions in L (G ) , and established the well-posedness of the Tricomi problem: consisting in finding
u u ( x, y ) which satisfies above mixed type equation in G and boundary condition:
0 on g1 ‰ g 2 .
a function
u
In 1993 , J. M. Rassias [44] considered the 2D elliptic and hyperbolic and parabolic partial differential
equation: Lu K ( y )u xx u yy r ( x, y )u f ( x, y ) , in a simply-connected domain G bounded by an
elliptic arc g1 ( for y ! 0) : connecting : A
Ac (1, 0)
(1, 0) ;
and
by two hyperbolic
characteristic curves:
g 2 ({ PA) : x 1 ³
y
K (t )dt ; g3c ({ AcP ) : x
0
intersecting at P { P (0, y p ),
y p ( 0) :
where
1 ³
y
0
³
yp
K (t )dt ( for y 0) ,
K (t ) dt
0
1 , as well as
by characteristic curves:
g 2c : x
³
y
0
K (t )dt ; g3 : x
³
y
0
K (t )dt ( for y 0) ,
2
proved a uniqueness theorem on quasi-regular solutions in C (G ) ˆ C (G ) and
the pertinent
2
existence theorem on weak solutions in L (G ) , and therefore he established the pertinent well-
u ( x, y ) which
0 on g1 ‰ g3 ‰ g3c .
posedness of solutions of the Tricomi problem: consisting in finding a function u
satisfies above mixed type equation in G and boundary condition: u
In 1997 and 1999, Rassias ([50]-[53]) introduced the elliptic-hyperbolic and parabolic equation:
Lu K1 ( y )u xx ( K 2 ( y )u y ) y r ( x, y )u f ( x, y ) , in a simply-connected domain G bounded by an
elliptic arc g1
( for y ! 0) : connecting : A (1, 0) ; Ac (1, 0) and by two characteristic curves:
g 2 ({ PA) : x 1 ³
y
K (t )dt ; g3 ({ AcP ) : x
0
intersecting at
P { P (0, y p ), where
1 ³
y p ( 0) :
2
y
0
³
yp
0
K (t )dt ( for y 0) ,
K (t ) dt
1 , proved a uniqueness
theorem on quasi-regular solutions in C (G ) ˆ C (G ) and the pertinent existence theorem on weak
2
solutions in L (G ) , and therefore, Rassias established the well-posedness of the Tricomi problem:
u ( x, y ) which satisfies above mixed type equation in G and boundary
0 on g1 ‰ g 2 .
consisting in finding
condition: u
u
Techniques to prove the existence of weak solutions :
Let us consider the relative adjoint equation:
L w
K1 ( y ) wxx ( K 2 ( y ) wy ) y r ( x, y ) w
and the adjoint boundary condition: w
f ( x, y ) ,
0 on g1 ‰ g3 .
Then Rassias introduced the following preliminary concepts:
90
International Journal of Applied Mathematics & Statistics
­°u ( p ) | p ( x, y )  G G ‰ wG : u u ( p ) ½°
®
¾;
°¯is twice continuously differentiable in G °¿
C 2 (G )
u
^
D
`
max DD u ( p) | p  G : D d 2 , such that
C 2 (G )
(D1 , D 2 ) : a1 , a2 ! 0, a
pD
xD1 yD 2 ,
D1
w
, D2
wx
p, p
xx yy ,
w
,
wy
( x, y )  \ 2 , and p
a1 a2 ; p
p
D u ( p)
p, p
( x , y )  \ 2 , and
1/ 2
;
( D1D1 D2D 2 u )( p ),
D
for sufficiently smooth functions ;
1/ 2
­
½
2
®u | ³ u ( p ) dp f ¾ ,
¯ G
¿
L2 (G )
dxdy ; D ( L)
with dp
u
^u  C
2
u
G
L2 ( G )
,
0 on g1 ‰ g 2 ` ,
(G ) : u
the domain of the formal operator L, and D ( L )
^w  C (G) : w 0 on g ‰ g ` ,
^u | D u (.)  L (G), D d 2` , and
2
1
D
2
§
·
2
¨ ³ u ( p ) dp ¸
©G
¹
the domain of the adjoint operator L ; W2 (G )
3
2
the complete normed Sobolev space with norm
u
u
2
(u
W22 ( G )
W22 (G, bd )
2
D ( L) .
L2 ( G )
D( L) .
W22 (G, bd )
^w  W
f ,w
¦ D u
L2 ( G )
D 2
§
D
¨¨ ¦ D u
© D d2
1/ 2
)
1/ 2
·
¸ ;
L2 ( G ) ¸
¹
2
: closure of D( L) with norm . 2 ;
2
W22 (G, bd )
2
2
2
Definition 2. 9.
2
D
: closure of D ( L ) with norm . 2 , or equivalently:
(G ) : Lu, w
u , L w
0
`
for all u  W22 (G, bd ) .
u ( p )  L2 (G ) is weak solution of Tricomi problem if
u
A function
0
u , L w , holds for all w  W22 (G, bd ).
0
0
Criterion 2. 10.
(i). A necessary and sufficient condition for the existence of a weak solution of Tricomi problem is that
the following a-priori estimate: w
0
d C L w ,holds for all w  W22 (G, bd ) for some positive
0
constant C. (ii). A sufficient condition for the existence of a weak solution of Tricomi problem is that
the following a-priori estimate: w 1 d C L w
2
0
,holds for all w  W2 (G, bd ) for some positive
constant C, where
1/ 2
w0
with
w
§
·
2
¨ ³ w( p ) dp ¸
©G
¹
L2 ( G )
w1t w
2
0
; w1
for all w  W2 (G, bd ) .
1/ 2
§
·
2
2
2
¨ ³ w( p ) wx ( p ) wy ( p ) dp ¸ ,
©G
¹
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
91
Rassias quoted that both the Hahn-Banach theorem and the Riesz representation theorem would
play an important role if above criterion 1 were not employed. For the justification of the definition of
weak solutions he applied Green’s theorem and classical techniques in order to show that:
f Lu and u 0 on g1 ‰ g 2 .
Rassias proved the following theorems for the well-posedness of Tricomi problem:
Existence Theorem 2. 11. Let us consider the Tricomi problem:
Lu
K1 ( y )u xx ( K 2 ( y )u y ) y r ( x, y )u
f ( x, y ) ; u
0 on g1 ‰ g 2 .
Also consider the above-mentioned simply-connected domain G and conditions:
R1 : r 0
on g 2 ; ( K1 K 2 )c ! 0 on g 2 ; K ic ! 0 (i 1, 2) in G ;
R2 : ( x c1 )dy ( y c2 )dx t 0 :
" star likedness " on g1 ;
­4(3r ( x c1 )rx ( y c2 )ry ) P1 d 4G11 0 for y t 0
( R3 ) : ®
d 4G12 0 for y d 0
¯ 4(2r ( x c1 )rx c2 ry ) P1
2
­ K ( y c2 ) K1c P 2 ( x c1 ) t G 21 ! 0 for y t 0
;
R4 : ® 1
c2 K1c P 2 ( x c1 ) 2
t G 22 ! 0 for y d 0
¯
­ K 2 ( y c2 ) K 2c P3 ( y c2 ) 2
2 K 2 c2 K 2c P3 (c2 ) 2
¯
R5 : ®
R6 : ³0
y
where
t G 31 ! 0 for y t 0
;
t G 32 ! 0 for y d 0
K (t )dt c2 K ( y ) c0 0 on g 2 ,
G ij ! 0 (i 1, 2,3; j 1, 2) ; K i (i 1, 2); r; f are sufficiently smooth; c1 1 c0 ,
and c0 ; c2 ; Pi (i
1, 2,3) are positive constants.
Then there exists a weak solution of the above pertinent Tricomi problem.
Uniqueness Theorem 2. 12. If one considers the mixed domain G,
the star-likedness condition:
( R1 ) : ( x 1)dy ydx t 0 on g1 ,
and the coefficient conditions:
­2r ( x 1)rx yry 0
( R2 ) : ®
¯ r ( x 1)rx 0
­ K1 ( y ) ! 0 whenever
° K (0) 0
°
( R3 ) : ® 1
° K1 ( y ) 0 whenever
°¯ K 2 ( y ) ! 0 everywhere
for y t 0
; r 0 on g 3 ;
for y d 0
y!0
y0
in
G
;
( R4 ) : K ic( y ) ! 0 (i 1, 2) in G ,
where K i (i
1, 2), and r are once-continuously differentiable and f is continuous, then this Tricomi
problem (T) has at most one quasi-regular solution in G.
92
International Journal of Applied Mathematics & Statistics
In 2002, Rassias [54] introduced the elliptic, bi-hyperbolic and bi-parabolic equation
K1 ( y )( M 2 ( x)u x ) x M 1 ( x)( K 2 ( y )u y ) y r ( x, y )u
Lu
f ( x, y ) ,
0, 0 y d 1 ; y
which is parabolic on both segments: x
0, 0 x d 1 ,
^ x, y  G( \ ) : x ! 0, y ! 0` , and hyperbolic in
^ x, y  G( \ ) : x ! 0, y 0` ; G ^ x, y  G( \ ) : x 0, y ! 0` ,
2
elliptic in the euclidean region: Ge
2
regions : Gh1
both
2
h2
with G the simply-connected mixed domain of our pertinent Tricomi problem.
In fact, Rassias considered:
f f ( x, y ) continuous in G, r r ( x, y ) once-continuously differentiable in G,
K i K i ( y ) (i 1, 2)
are
once-continuously
differentiable
for
y  > k1 , k2 @
:
inf ^ y : ( x, y )  G` ; k2
k1
sup ^ y : ( x, y )  G` ,and M i
M i ( x) (i 1, 2)
are once-continuously differentiable for
x  > m1 , m2 @ : m1
inf ^ x : ( x, y )  G` ; m2
sup ^ x : ( x, y )  G` .
Besides he assumed the following conditions:
­! 0
°
K1 ( y ) ® 0
° 0
¯
K1
as well as : K 2
K
K ( y)
y!0
y 0 and
y0
for
for
for
K 2 ( y ) ! 0 and
­! 0
K1 ( y ) °
® 0
K 2 ( y) °
¯ 0
for
for
for
y!0
y 0 and M
y0
and
y o0
­! 0
°
K ( y ) M ( x) ® 0
° 0
¯
for
for
for
for
for
for
x!0
x 0 ,
x0
M 2 ( x) ! 0 everywhere in G ,so that
M2
Also Rassias supposed that : lim K ( y )
KM
­! 0
°
M 1 ( x) ® 0
° 0
¯
M1
­! 0
M 1 ( x) °
® 0
M 2 ( x) °
¯ 0
M ( x)
for
for
for
x!0
x 0 .
x0
lim M ( x) exist in G ;
x o0
x ! 0, y ! 0
0; y 0
x
.
x ! 0, y 0; x 0, y ! 0
Rassias quoted that it is not considered here: KM ! 0 for x 0, y 0 , and the above mixed
type equation degenerates its order at the origin O(0,0). Besides Rassias considered the boundary
wG of the domain G formed by curves: A curve g1 which is the elliptic arc lying in the first quadrant:
x ! 0, y ! 0 and connecting the points A(1, 0) and B(0,1) ;two characteristic arcs
g2 : ³
x
1
M (t ) dt
³
y
0
K (t ) dt , g3 : ³
x
0
M (t )dt
³
y
0
K (t )dt ,
descending from the points A(1, 0) and O (0, 0) until they terminate at a common point of
intersection P1 ( x p1 , y p1 ) in the fourth quadrant :
J 2 and J 3 :
x ! 0, y 0 ; and two hyperbolic characteristic arcs
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
J2 : ³
x
0
M (t )dt
³
y
1
K (t )dt , J 3 : ³
x
M (t )dt
0
³
y
0
93
K (t )dt ,
emanating from the points %(0,1) and O (0, 0) until they terminate at a common point of
intersection P2 ( x p2 , y p2 ) in the second quadrant :
x 0, y ! 0 . Rassias assumed the boundary
0 on g1 ‰ g 2 ‰ J 2 , and suggested the following Problem (T): The Tricomi problem,
u u ( x, y ) which satisfies the above equation in G and
the boundary condition on g1 ‰ g 2 ‰ J 2 which is a “continuous” part of the boundary wG of G.
condition: u
or Problem (T) consists in finding a function
Definition 2. 13. A function
2
(i) u  C (G ) ˆ C (G ),
u
u ( x, y ) is a quasi-regular solution of Problem (T) if :
G ‰ wG;
G
(ii) the Green’s theorem (of the integral calculus) is applicable to the integrals
³³ u Ludxdy, ³³ u Ludxdy ;
x
G
y
G
(iii) the boundary and region integrals, which arise, exist;
(iv) u satisfies the above mixed type equation in G and the boundary condition on g1 ‰ g 2 ‰ J 2 .Then
Rassias stated and proved:
Uniqueness Theorem 2. 14. If one assumes the above conditions in the mixed domain G, and the
star-likedness condition: xdy ydx t 0 on g1 , and conditions:
­2r xrx yry 0 for x t 0, y t 0
°
; r 0 on g 3 ‰ J 3 ;
® r xrx 0 for x t 0, y d 0
° r yr 0 for x d 0, y t 0
y
¯
­ K 2 ( y ) yK 2c ( y ) ! 0 for y t 0
°
,
® M 2 ( x) xM 2c ( x) ! 0 for x t 0
° K c( y ) ! 0
c
in
G ; M i ( x) ! 0 in G , for i 1, 2
i
¯
then this Tricomi problem (T) has at most one quasi-regular solution in G.
Case K1
sgn( y ) ; M 1
sgn( x) ; K 2
M2
1 : was investigated (in1984) [32].
3. THE TRICOMI-PROTTER PROBLEM
In 2007, Rassias [55] introduced the nD ( n ! 2) ) parabolic elliptic- hyperbolic- parabolic partial
differential equation
n
Lu { K1 (t )(¦ u xi xi ) ( K 2 (t )ut )t r ( x, t )u
f ( x, t )
(3.1)
i 1
( x1 , x2 ,..., xn ) ,
and K 2 (0) ! 0 ; K 2 (t ) ! 0 for all real t z 0 , as well as K1 (0) 0 ; K1 (t ) ! 0 for t ! 0 ;and
K1 (t ) 0 for t 0 ,as well as ( )t w / wt .
which is parabolic on
t
0 ; elliptic in t ! 0 ;and hyperbolic in t 0 , for x
Besides Rassias investigated the pertinent Tricomi - Protter problem of this equation. Furthermore he
established uniqueness of quasi-regular solutions for the afore-mentioned Tricomi – Protter problem.
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International Journal of Applied Mathematics & Statistics
Rassias introduced the parabolic elliptic-hyperbolic equation (3.1) in a bounded simply-connected
mixed domain G with a piecewise smooth boundary
continuous in G,
S1 ‰ S3 ‰ S4 , where f
r ( x, t ) is once-continuously differentiable in G, K i
r
monotone increasing continuously differentiable for
sup{t : ( x, t )  G} , for x
k2
wG
t  [k1 , k2 ] with k1
( x1 , x2 ,..., xn ) . Let us denote K1 (t )
that the boundary wG of a bounded simply-connected region G of
following surfaces: A piecewise smooth surface
G
S3 , S4 of (3.1): S3 : < ( x, t ) { x 1 U
G
the hyperbolic region
n
i 1
where
x
¦x
2
i
inf{t : ( x, t )  G} and
K (t ) K 2 (t ) . Rassias quoted
\ n 1 (n ! 2) is formed by the
{( x, t )  G : t
0} in x
0 , and S 4 : ) ( x, t ) { x U
1 ;two
0 , lying in
{( x, t )  G : t 0} and satisfying the conditions
K1 (t )[¦ (w< / wxi ) 2 ] K 2 (t )(w< / wt ) 2 |S3 0
n
K i (t ) (i 1, 2) are
S1 : F ( x, t ) 0 lying in the elliptic region
{( x, t )  G : t ! 0} which intersects the parabolic plane G0
characteristics
f ( x, t ) is
and
U
i 1
³
t
0
n
;
K1 (t )[¦ (w) / wxi ) 2 ] K 2 (t )(w) / wt ) 2 |S4 0 ,
i 1
K (W ) dW (! 0) with
K (W )
K1 (W ) / K 2 (W ) 0 for
K1 (W ) 0 and K 2 (W ) ! 0 if W 0 . Besides S3 ascends from a fixed point P (0, t p ), t p 0 and
intersects the plane
intersects
G0 in x
1 for t 0 , and S4 descends from the fixed origin O(0,0), and
S3 for t 0 : t ! t p . Furthermore if G is bounded, instead of S3 , by a piecewise smooth
non-characteristic surface
S3c : 4( x, t ) 0 which intersects the plane G0 in x
1 and satisfies the
n
non-negative condition :
K1 (t )[¦ (w4 / wxi ) 2 ] K 2 (t )(w4 / wt )2 |S3c t 0 ,then the surface S3c lies
i 1
inside the characteristic 3D-space bounded by the characteristic surfaces
S3 and S4 of (3.1). Let us
assume the boundary condition
u
If
0 on S1 ‰ S3 .
(3.2)
S3 is replaced by S3c , then the boundary condition (3.2) is replaced accordingly. Rassias named
the following problem after Protter (1977), because Protter was the pioneer investigator for a class of
Goursat type problems for the wave equation in dimensions greater than 2 , fundamental for the
Tricomi problem, and his results today consist a cornerstone in boundary value problems of mixed
type partial differential equations.
According to Protter ([Bull. Amer.Math. Soc. ,1 (3) (1979), 534-538] ; [25] ):
“very little is yet known of the nature of solutions of equations of mixed type when the number of
independent variables is more than two, and that the task of studying mixed type boundary value
problems in three and more dimensions appears formidable and more remote.”
Rassias ([20], [23-26], [36]) generalized the Tricomi and Frankl problems in n ( ! 2) dimensions
based on Protter’s original proposal to him. This generalization in multi-dimensional regions was a
breakthrough for the Tricomi and Frankl problems in domains of higher than two dimensions for the
generalized multi-dimensional Chaplygin equation with more general coefficients. Rassias now
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
95
generalizes even further his own results obtained in his Ph. D thesis (U. C., Berkeley, 1977),
through the following investigation:
The Tricomi - Protter problem, or Problem (TP) consists in finding a function u u ( x, t ) which
satisfies the mixed type partial differential Eq. (3.1) in G and the boundary condition (3.2) on the
surface portion S1 ‰ S3 of the boundary wG of G.
If
S3 is replaced by S3c , the Tricomi - Protter problem, or Problem (TP) is recalled accordingly as
the Frankl - Protter problem, or Problem (FP) .
Definition 3. 1. A function
conditions hold:
u
u ( x, t ) is a quasi-regular solution of Problem (TP) if the following two
(i) the Green’s theorem (of the integral calculus) is applicable to the integrals
³³ u Lu dxdt , ³³ u
xi
Lu dxdt (i 1, 2,..., n) and
G
G
³³ u
t
Lu dxdt ;
G
(ii) the boundary surface and region integrals, which arise, exist; and (iii) u satisfies the mixed type
Eq. (3.1) in G and the boundary condition (3.2) on wG S1 ‰ S3 ‰ S 4 .
Besides Rassias stated and proved the following general nD uniqueness theorem:
Uniqueness Theorem 3. 2. Consider the parabolic elliptic- hyperbolic Eq. (3.1) and the boundary
condition (3.2), as well as the afore-described bounded simply-connected mixed domain G of the (x,
\ n 1 (n ! 2) . Besides assume that K 2 (0) ! 0 ; K 2 (t ) ! 0 for all real t z 0 , as
well as K1 (0) 0 ; K1 (t ) ! 0 for t ! 0 ;and K1 (t ) 0 for t 0 .Furthermore suppose the conditions:
t) - euclidean space
( R1 ) : The piecewise smooth surface S1 : F ( x, t ) 0 is ” strongly star-like”,
n
¦ xQ
such that
i i
i 1
vector on
wG
c(t )Q n 1 |S1 t 0 for t ! 0 ,where Q
(Q 1 ,Q 2 ,...,Q n ,Q n 1 ) is the normal unit
S1 ‰ S3 ‰ S4 , and
t
³
c(t )
K (W ) dW / K (t ) ! 0 for t ! 0 , K (t )
K1 (t ) / K 2 (t ) ! 0 ,
0
with characteristic surfaces
S3 and S4 of equation (3.1) described above ;
n
(2a nr ) ¦ xi rxi (cr )t ( K 2 ac)c t 0 in G,
( R2 ) :
i 1
a [(n 1) / 2] [( K1 (t ) K 2 (t ))cc / 4 K1 (t ) K 2 (t )]
[(n 1) / 2] {[[( K (t ))c / K (t )] 2[( K 2 (t ))c / K 2 (t )]](c / 4)} ,
t
c
³
| K (W ) | dW / | K (t ) |
0
in G and
( )c d ( ) / dt ;
( R3 ):
R (t ) (aK 2 K U n 1 )c acK 2 K U n 1 0 for t 0 ,
U
³
t
0
K (W ) dW (! 0) for t 0 ;
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International Journal of Applied Mathematics & Statistics
R* (t )
( R4 ) :
K 2 U n 2(Tn21 (t ) / Tnc1 (t ) ! 0 for t 0 ,
with a real valued function
Tn 1
Tn 1 : \ o \ (n t 2), such that
Tn 1 (t ) o 0 as t o 0 and Tnc1
dTn 1 (t ) / dt
R(t ) |S4 (! 0) ,
f f ( x, t ) is continuous in G, r r ( x, t ) is once-continuously differentiable in G,
K i K i (t ) (i 1, 2) are monotone increasing
continuously differentiable for t  [ k1 , k2 ]
with : k1 inf{t : ( x, t )  G} , k2 sup{t : ( x, t )  G} , for x ( x1 , x2 ,..., xn ) .
and
Then the above Problem (TP) has at most one quasi-regular solution in G.
Proof . We apply the well known
a, bi (i 1, 2,..., n), c energy integral method with choices in G:
t
a [(n 1) / 2] [( K1 (t ) K 2 (t ))cc / 4 K1 (t ) K 2 (t )] , bi
xi , c
³
| K (W ) | dW / | K (t ) |
0
for all
i 1, 2,..., n . Then we use the above mixed type Eq. (3.1) as well as the boundary condition
u1 , u2 of the Problem (TP). Then, we claim that u u1 u2 0
(3.2). We assume two solutions
holds in the domain G. In fact, we investigate
0
J
2(lu, Lu )0
³³ 2 lu Lu dxdt
(3.3)
G
n
a (t )u ¦ bi ( xi )u xi c(t )ut ; a
lu
where
a (t ), bi
bi ( xi ), c
c(t ) are defined above. Thus
i 1
Lu
Lu1 Lu2
ff
0 in G. We introduce the new differential identities
(2aK1uu xi ) xi 2aK1u x2i ((aK1 ) xi u 2 ) xi ( K1a) xi xi u 2
2aK1uu xi xi
(2aK1uu xi ) xi 2aK1u x2i ;
(2aK 2uut )t 2aK 2ut2 ((aK 2 )cu 2 )t ( K 2 a )ccu 2 ,
2aK 2uutt
2aruu
2aru 2 ; 2aK 2cuut
(aK 2cu 2 )t (aK 2c )cu 2 ;
(2bi K1u xi u x j ) x j (bi K1u x2j ) xi (bi K1 ) xi u x2j 2(bi K1 ) x j u xi u x j
2bi K1u xi u x jj
(2bi K1u xi u x j ) x j (bi K1u x2j ) xi (bi K1 ) xi u x2j (i z j : i, j : 1, 2,..., n);
2bi K1u xi u xi xi
2bi K 2u xi utt
2bi ru xi u
2cK1u xi xi ut
(bi K1u x2i ) xi (bi K1 ) xi u x2i ;
(2bi K 2u xi ut )t (bi K 2ut2 ) xi (bi K 2 ) xi ut2 2(bi K 2 )t u xi ut ;
(bi ru 2 ) xi (bi r ) xi u 2 ; 2bi K 2cu xi ut
2(bi K 2 )t u xi u;
(2cK1u xi ut ) xi (cK1u x2i )t (cK1 )cu x2i 2(cK1 ) xi u xi ut
(2cK1u xi ut ) xi (cK1u x2i )t (cK1 )cu x2i ; 2cK 2ut utt
2crut u
(cru 2 )t (cr )t u 2 ; 2cK 2cut utt
(cK 2ut2 )t (cK 2 )cut2 ;
(cK 2cut2 )t (cK 2c )cut2 .
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
97
We note that:
n
¦ (bQ
i
j
iz j
n
2¦ (biQ j b jQ i )u xi u x j
b jQ i )u xi u x j
i j
n
2¦ (biQ j )u xi u x j .
iz j
Furthermore we employ the classical Green’s theorem of the integral calculus. Therefore if
dxdt , and dS the surface element ,we get the fundamental identity
dV
0=J= 2
n
[a (t )u ¦ bi ( xi )u xi c(t )ut ]Lu dV
³³
i 1
G
n
=
³³ [2ar ¦ (b r )
i
i 1
G
n
+
i 1
(bi K1 ) xi ¦ (b j K1 ) x j (cK1 )c]u x2i
1
i 1
+ [ 2aK 2
(cr )t ¦ ( K1a) xi xi ( K 2 ac)c]u 2 dV
n
³³ {¦ [2aK
G
n
xi
j zi
n
¦ (bi K 2 ) xi (cK 2 )c 2cK 2c ]ut2
i 1
-2
n
n
¦ (K b )
u xi u x j 2¦ ( K1c) xi u xi ut }dV
1 i xj
iz j
i 1
n
v³ (¦ bQ
+
i i
i 1
wG
cQ n 1 )r u 2 dS
n
+
v³ {2au ( K ¦ u Q
1
xi
i 1
wG
n
+
n
i
K 2utQ n 1 ) u 2 [¦ ( K1a ) xi Q i acK 2Q n 1 ]}dS
i 1
n
v³ [¦ (b Q ¦ bQ
j
j 1
wG
i i
iz j
n
+2
j
¦ (bQ
i
i j
n
cQ n 1 ) K1u x2j (¦ biQ i cQ n 1 ) K 2ut2
i 1
n
j
b jQ i ) K1u xi u x j 2¦ (bi K 2Q n 1 cK1Q i )u xi ut ]dS
i 1
= I1 I 2
J1 J 2 J 3 .
(3.4)
n
2aK1 (bi K1 ) xi ¦ (b j K1 ) x j (cK1 )c 0 for i 1, 2,..., n ,
We note that: Ai
j zi
n
B
2aK 2 ¦ (bi K 2 ) xi (cK 2 )c 2cK 2c
0.
i 1
n
Therefore
I2
³³ (¦ Au
G
From condition
i 1
2
i xi
But2 )dV
( R2 ) ,one gets I1
0 , because Ai
n
³³ [(2a nr ) ¦ x r
G
We claim that
J1
Bi
0:
In fact, this is valid from condition (3.2) because
i 1
i xi
( K1bi ) x j
( K1c) xi
0.
(cr )t ( K 2 ac)c]u 2 dV t 0 .
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International Journal of Applied Mathematics & Statistics
Q |S
(Q 1 ,Q 2 ,...,Q n ,Q n 1 )
4
’) / ’)
( x1 , x2 ,..., xn , N ) K / N 1 K |S4 ,
t
N
such that
K ( ³ K (W )dW )(! 0) for t 0 ,and
U K
0
’) |S4 () x1 , ) x2 ,..., ) xn , ) t ) ( x1 / U , x2 / U ,..., xn / U , K ) , ’)
1 . ,
n
characteristic equation:
K1 (t )[¦ (w) / wxi ) 2 ] K 2 (t )(w) / wt ) 2 |S4 0 holds, and
i 1
n
n
t
i 1
i 1
0
[¦ biQ i cQ n 1 ] |S4 ( K / N 1 K ){¦ xi2 [ N ³ K (W )dW / K (t )]}|S4 0 .
The rest of the proof is based on analogous techniques and the application of the classical CauchySchwarz-Buniakowski inequality.
We are now investigating
J 2 : In fact, let us note that S4 :
U cos T cos M1 cos M2 ...cos Mn 3 cos Mn 2 ;
U sin M1 cos M2 ...cos Mn 3 cos Mn 2 ;
x1
x3
x2
U sin T cos M1 cos M2 ...cos Mn 3 cos Mn 2 ;
x4 U sin M 2 ...cos M n 3 cos M n 2 ;
………………………………
U sin Mn 4 cos Mn 3 cos Mn 2 ; xn 1
xn 2
for
U sin Mn 3 cos Mn 2 ; xn
U sin Mn 2 ; t t ,
T  [0, 2S ), M j  [S / 2, S / 2] ( j 1, 2,..., n 2); xi  \; and t  (f, 0] .
u (t ,T , M1 , M 2 ,..., Mn 2 ) u ( x1 , x2 ,..., xn , t ) |S4
We consider:
u ( U cos T cos M1...cos Mn 2 , U sin T cos M1...cos M n 2 ,..., U sin Mn 2 , t ) .
du |S4
Thus
n
¦u
i 1
xi
dxi ut dt
j 1
d U (t ) / K (t ) . Let us denote
dt
Thus we obtain dxi |S4
wu / wt
n2
ut dt uT dT ¦ uM j dM j ,such that U
t
³ K (W ) dW yielding
0
fi (t ,T , M1 ,..., Mn 2 ) .
xi |S4
n2
( f i )t dt ( fi )T dT ¦ ( f j )M j dM j and
j 1
[ K (u x1 cos T cos M1...cos M n 2 u x2 sin T cos M1...cos M n 2 ...
u xn sin Mn 2 ) ut ] |S4 wu / wT |S4 wu / wT |S4 ,
which is the derivative in the direction of the tangent vector
T
(t1 , t2 ,..., tn , tn 1 ) (wx1 / wt , wx2 / wt ,..., wxn / wt , wt / wt )
K ( x1 , x2 ,..., xn , U / K ) / U ,
of u in the direction of one of the generators of
Q xT
n 1
¦Q t
i i
i 1
n
S4 ,such that the dot (x) product
[¦ ) xi (wxi / wt ) ) t (wt / wt )] 0 . Therefore,
i 1
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
n
K1 ¦ u xiQ i K 2utQ n 1 |S4
99
K 2 K /(1 K )[ K (u x1 cos T cos M1...cos M n 2
i 1
(3.5)
... u xn sin Mn 2 ) ut ] K 2 K /(1 K ) (wu / wt )
where
cos T cos M1...cos M n 2 = x1 / U
dx |S4 Q n 1dS
But
( 1 K / K ) J dM dT dt ,
( 1 K / K )dx
n
n2
i 1
j 1
3 dxi , dM
dx
1 K Q n .
() t / ’) )dS . Hence
dS |S4 ( ’) / ) t ]dx
where
1 K Q 1 ,…, sin M n 2
3 dM j ; and
J (t , M1 , M 2 ,..., Mn 2 ) =
J
t
K ( ³ K (W ) dW ) n 1 C (M )
w ( x1 , x2 ,..., xn 1 , xn ) / w (T , M1 ,..., Mn 2 , t )
U n 1 K C (M ), where
0
n2
C (M )
for
3 cos M j
j
j 1
(cos M n 2 )
n2
... cos M 2 cos M1 (! 0) ,
2
M j  [S / 2, S / 2] ( j 1, 2,..., n 2)
We note that
C (M ) 1, for n 1 and n
dS |S4
U
for
n ! 2 and J the Jacobian.
2 . Therefore
n 1
1 K C (M )dM dT dt .
(3.6)
Therefore from (3.5)-(3.6), one gets
n
( K1 ¦ u xiQ i K 2utQ n 1 )dS |S4 U n 1 K 2 K C (M )(wu / wt )dM dT dt .
(3.7)
i 1
If
F ( x, t ) : \ n 1 o \ is a given real function, we find
F
J 2*
³ F ( x, t )dS ³ F ( x, t ( x))( ’) / ) )dx,
t
S4
S4c is the projection of S4 into the x-space, such that
where
x
n
t
i 1
0
(¦ xi2 )1/ 2 ; and x ³ K (W )dW
Let us denote
64
S4c
M
0 (or t
t ( x) ).
(M1 , M 2 ,..., Mn 2 ) . Thus from (3.6) and (3.8) and denoting
{(T , M , t )  \ n 1 : 0 d T d S , S / 2 d M j d S / 2( j 1, 2,..., n 2), t * (T , M ) d t 0} ,
we get
t
J 2*
³ F (³ K (W )dW cos T cos M1...cos Mn2 ,..., t ) J 1 K / K dtdM dT
64
0
t
³ F ( ³
64
0
K (W )dW cos T cos M1...cos M n 2 ,..., t ) U n 1 (t )C (M )dtdM dT
(3.8)
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International Journal of Applied Mathematics & Statistics
We note that
6 4 is the region in the TM t space into which the region S4c is mapped under the
following transformation
U cos T cos M1 cos M2 ...cos Mn 3 cos M n 2 ; …; xn
x1
U sin Mn 2 .
(3.9)
Let us denote
6c4
{(T , M , U )  \ n 1 : 0 d T d S , S / 2 d M j d S / 2( j 1, 2,..., n 2), 0 d U d R(T , M )}
the region in the
TMU space
S4c is mapped under the transformation
into which either the region
6 4 is mapped under transformation
(3.9) or the region
t
U
³ K (W ) dW .
(3.10)
0
In this latter case the Jacobian is
wt / wU 1 and d U / dt
K (t ) .
(3.11)
(1/ K )d U dM dT .
(3.12)
Thus
dtdM dT
From the geometry of
Q n 1dS |S
dt / d U d U dM dT
S4 ,we obtain
( K / 1 K ) U n 1 1 K C (M )dT dM dt
4
K U n 1 C (M )dT dM dt .
Therefore from the boundary condition (3.2), one proves
J2
2
³ 2D uU n 1 K 2 K (wu / wt )C (M )dT dM dt ³ u acK 2 U n 1 KC (M )dT dM dt .
64
64
From this and integrating by parts and by virtue of the fact that
S4 , we get
the upper and lower limits of
³ (u) [(D K
K U n 1 )c acK 2 K U n 1 ]C (M )dT dM dt .
2
J2
lim K (t ) 0 and that u vanishes at
t o 0
2
(3.13)
64
We now investigate
J3 :
In fact, from condition
³
J3
( R1 ) and from the boundary condition (3.2) we prove that
³
Q(u x1 ,u x2 ,..., u xn , ut )dS
S1 ‰ S3 ‰ S4
³
S1 ‰ S3
S1 ‰ S3
n
( N ) [ K1 ¦Q K Q
* 2
2
i
2
2 n 1
i 1
Q (u x1 ,u x2 ,..., u xn , ut )dS ³ Q(u x1 ,u x2 ,..., u xn , ut )dS
S4
n
][¦ xiQ i c(t )Q n 1 ]dS 2 ³ (wu / wt ) 2 ( K 2 / 1 K ) U dS
i 1
S4
t 2 ³ (wu / wt ) ( K 2 / 1 K ) U dS
2
S4
³
= 2 (wu / wt ) ( K 2 U )C (M ) dT dM dt
64
2
n
(3.14)
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
N * is the normalizing factor, such that
where
n
¦u
0 du |S1 ‰ S3
i 1
n
and
¦Q
[ K1
i 1
2
i
xi
* 2
= (N ) [
¦ xQ
i i
i 1
N *Q n 1 ,
n
i 1
Q (u x1 ,..., u xn , ut ) |S1 ‰ S3 ‰ S4
n
u |S1 0 implies
K 2Q n21 ]| S1 t 0 and characteristic equation [ K1 ¦Q i2 K 2Q n21 ] |S3 0 ,
as well as the quadratic form Q on the boundary wG
Q
boundary condition
N *Q i (i 1, 2,..., n), ut
dui ut dt ; u xi
101
n
¦Au
j 1
* 2
j xj
S1 ‰ S3 ‰ S 4 :
n
n
i j
i 1
Bt*ut2 ¦ Aij*u xi u x j 2¦ Bi*u xi ut
n
c(t )Q n 1 ][ K1 ¦Q i2 K 2Q n21 ] |S1 ‰ S3 2( K 2 / 1 K ) U (wu / wt ) 2 |S4 ,
i 1
n
n
t ( N * ) 2 [¦ xiQ i c(t )Q n 1 ][ K1 ¦Q i2 K 2Q n21 ] |S1 2( K 2 / 1 K ) U (wu / wt ) 2 |S4
i 1
i 1
2( K 2 / 1 K ) U (wu / wt ) |S4 ( t 0) ,
2
where on S 4 :
A*j
n
n
¦ (b Q ¦ bQ
j
j
i i
j 1
Bt*
Aij*
iz j
n
t
i 1
0
(¦ biQ i cQ n 1 ) K 2 |S4 = 2( K 2 K / N 1 K )( ³ K (W )dW ) 2 t 0 ,
n
¦ (bQ
i
j
i j
Bi*
b jQ i ) K1 |S4 = 2( KK 2 K / N 1 K ) xi x j (i j; i, j : 1, 2,..., n) ,
n
¦ (b K Q
2 n 1
i
i 1
cK1Q i ) |S4 = 2( K 2 K / 1 K ) xi (i 1, 2,..., n) .
0 , we obtain
Therefore from J1
J4
cQ n 1 ) K1 |S4 = 2 KK 2 ( K / N 1 K ) x 2j t 0( j 1, 2,..., n),
J1 J 2 J 3 t ³ {(u ) 2 [(aK 2 K U n 1 )c acK 2 K U n 1 ]
64
2(ut ) ( K 2 U )}C (M )dT dM dt
2
n
where ut
wu / wt .
Assuming
a
t o 0 ; Tnc1
real
valued
dTn 1 (t ) / dt
function Tn 1 : \ o \ ( n t 2),
[(D K 2 K U
n 1
)c acK 2 K U
such
n 1
that Tn 1
Tn 1 (t ) o 0 as
] |S4 R (t ) |S4 (! 0)
from condition ( R3 ) .From integration by parts and by virtue of the fact that lim K (t )
t o0
u vanishes at the upper and lower limits of S 4 ,
one gets that
³ Tc
n 1
64
(t )(u ) 2 C (M )dT dM dt
2 ³ Tn 1 (t )u ut C (M )dtdM dT .
64
0 and that
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International Journal of Applied Mathematics & Statistics
From this and Cauchy-Schwarz-Buniakowski inequality, as well as: Tnc1 (t ) ! 0 on S 4 (t 0) and if
Tn 1 / Tnc1 is integrable, one gets on S4 that
³ 7c
n 1
³ 7cn 1 (t )(u ) 2 C (M )dT dM dt
(t )(u ) 2 C (M )dT dM dt
64
2
64
³T
n 1
(t ) u ut C (M )dtdM dT
64
2 ³ [ Tnc1 (t ) u ][(Tn 1 (t ) / Tnc1 (t )) ut ]C (M ) dtdM dT
d 2[ ³ 7cn 1 (t )(u ) 2 C (M ) dT dM dt ]1/ 2 [ ³ (72n 1 (t ) / Tnc1 (t ))(ut ) 2 C (M )dT dM dt ]1/ 2 ,
64
64
or
³ 7c
n 1
64
(t )(u ) 2 C (M ) dT dM dt d 4 ³ (72n 1 (t ) / Tnc1 (t ))(ut ) 2 C (M )dT dM dt .
64
Therefore from this inequality we find :
³ [2K U
J4
2
n
(ut ) 2 Tnc1 (t )(u ) 2 ]C (M )dT dM dt
64
t 2 ³ [. 2 U n 2(Tn21 (t ) / Tnc1 (t )](ut ) 2 C (M )dT dM dt (! 0) ,
64
as
[ K2
U n 2(Tn21 (t ) / Tnc1 (t ) ]| 6
4
=
{[. 2 U nTnc1 (t ) 2Tn21 (t )] / Tnc1 (t )}|64 ! 0 ,
or
Tnc1 (t ) R* (t )
K 2 U nTnc1 2Tn21 ! 0 for t 0 ,
which holds by condition ( R4 ) ,and the proof of our above theorem is complete.
NOTE: One could easily prove the existence of weak solutions and thus the well-posedness of the
Tricomi-Protter problem via well-known techniques ([29],[53]).
Remarks 3. 3.
(i). The author observed that employing a variation of the a, bi (i
method, one obtains the above sufficient conditions ( Ri ) (i
1, 2,..., n), c energy integral
1, 2,3) for the uniqueness of the quasi-
regular solutions of the Tricomi - Protter boundary value problem (3.1) and (3.2).
(ii). If one takes K1
K , K2
1 in this theorem and follows Rassias proof, one establishes the
uniqueness result of Rassias Ph. D. dissertation (1977) [20].
(iii). The following case :
K 2 (0) 0 ; K 2 (t ) ! 0 for all real t z 0 , as well as K1 (0) 0 ; K1 (t ) ! 0 for t ! 0 ;and K1 (t ) 0
0 , is analogous in
for t 0 , where the order of the mixed type equation (3.1) is degenerated at t
investigating pertinent quasi-regular solutions of (3.1).
In 2007, J. M. Rassias and A. Hasanov [56] considered the elliptic type equation with singular
coefficient:
Int. J. Appl. Math. Stat.; Vol. 13, No. J08, June 2008
LOD u { u xx u yy u zz in the domain D
2D
u x O 2u
x
0, 0 2D 1, O
103
O1 iO2 , O1 , O2  \,
^ x, y, z : 0 x, f y f, f z f` .
Then they found fundamental solutions of the above equation , expressed through confluent
hypergeometric functions of Kummer H 3 a, b; c; x, y from two arguments. Also they proved by
means of expansion confluent hypergeometric functions of Kummer, that the constructed solutions
have a singularity of the order 1/ r at r o 0 .Besides, if :
O2
P 2 ,they solved some boundary
value problems in domain D .
In 2007, A. Hasanov [10] considered the generalized Rassias bi-hyperbolic equation:
R (u ) { y m z k u xx x n z k u yy x n y mu zz
3
in the field of : \ 0; m, n, k : costant (! 0) ,
^ x, y, z : x ! 0, y ! 0, z ! 0` .
By means of a suitable change of variables he reduced this generalized Rassias equation to a system
of hypergeometric equations for the function of Lauricella of three variables and found eight linearly
independent particular solutions of the said system of hypergeometric equations. Besides he found
properties of these particular solutions by virtue of decomposition of the hypergeometric function of
Lauricella.
In 2007, J. M. Rassias and G. Wen [57] focused their investigation on the oblique derivative problem
for general second order mixed type equations with a nonsmooth parabolic degenerate line, which
included the famous Tricomi problem as a special case. First, they gave the formulation of the above
problem, and then proved the solvability of the pertinent problem for the mixed equations with
nonsmooth degenerate line.
Besides they introduced a new notation, such that the second order equation of mixed type could be
reduced to the mixed complex equation of first order and then they used the advantage of the
complex analytic method ; otherwise, this method could not be employed.
In 2007, G. Wen, D. Chen and X. Cheng [65] investigated:
The General Tricomi-Rassias (or GTR) problem for the generalized Chaplygin equation. This is one
general oblique derivative problem that includes: The exterior Tricomi-Rassias Problem as a special
case. The investigation of these results are established by the employment of a new method. In 2007,
G. Wen [66] has recently written an excellent book in mixed type equations.
Additional references are the following: ([3],[6],[11-13],[15],[18],[29-30],[34-35],[38],[42],[45-49],[5152],[58],[61]).
4. Open Problems on Mixed Type Equations and Systems
Let us suggest the following twelve open problems:
Problem 4. 1. An open question concerns the regularity of solutions for the boundary value problems
of mixed type in the sense of Tricomi and Frankl.
Problem 4. 2. An open question remains of proving the well-posedness of regular solutions without
restrictions on the functional coefficients or the geometrical size or the shape of the mixed domain.
Problem 4. 3. No serious work has been established on nonlinear initial and boundary value
problems of mixed type in three and more dimensions.
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International Journal of Applied Mathematics & Statistics
Problem 4. 4. The problem concerning the solution of elliptic systems in a domain on the boundary
of which the type degenerates is not sufficiently investigated.
Problem 4. 5. Little is known about the Cauchy problem for hyperbolic equations of order higher
than two with boundary conditions on the curve of parabolic degeneracy.
Problem 4. 6. The correct statement of the problem for equations of mixed type in multi-dimensional
regions still remains to be a very intricate mathematical problem.
Problem 4. 7. The study of higher order partial differential equations and systems of partial
differential equations of mixed type requires much more investigation.
Problem 4. 8. It would be very interesting to clear up the question whether there is an extremal
principle for the initial and boundary value problems of mixed type.
Problem 4. 9. One of the most important problems of mathematical physics is the
investigation of the properties of solutions of partial differential equations of mixed type with boundary
conditions in the sense of Tricomi and Frankl.
Problem 4. 10. To consider regions of mixed type multi-connected with parabolic lines of degeneracy
replaced by arbitrary curves (not necessarily straight lines) is a question of a very high demand and
difficult to handle.
Problem 4. 11. The existence problem for regular transonic flow around given general profile with
given velocity at f is very difficult to be solved completely.
Problem 4. 12. To solve the Tricomi, Frankl and Bitsadze-Lavrent’ev problems for any one of the
following eight pertinent mixed type partial differential equations:
( y x)u xx u yy r ( x, y )u
( y x 2 )u xx u yy r ( x, y )u
f ( x, y ); ( x 2 y 2 1)u xx u yy r ( x, y )u
f ( x, y ); ( x 2 y 2 1)u xx u yy r ( x, y )u
z (u xx u yy ) xu zz r ( x, y, z )u
f ( x, y ) ;
f ( x, y ) ;
f ( x, y, z ); z (u xx u yy ) xu zz r ( x, y, z )u
( z y )u xx ( z x)u yy u zz r ( x, y, z )u
f ( x, y , z ) ;
f ( x, y, z );
sgn( z y )u xx sgn( z x)u yy u zz r ( x, y, z )u
f ( x, y , z ) .
ACKNOWLEDGEMENT
I am deeply grateful to the three (3) Reviewers of this paper for their precious suggestions.
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