10_xxx_rassias.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 4, No. 2, 2011, 186-208

ISSN 1307-5543 – www.ejpam.com

The Exterior Tricomi and Frankl Problems for Quaterelliptic-

Quaterhyperbolic Equations With Eight Parabolic Lines

John Michael Rassias

National and Capodistrian University of Athens, Pedagogical Department, Section of Mathematics

and Informatics, 4, Agamemnonos Str., Aghia Paraskevi Athens, Attikis 15342, Greece

Abstract. The famous Tricomi equation was established in 1923 by F. G. Tricomi who is the pioneer

of parabolic elliptic and hyperbolic boundary value problems and related problems of variable type. In

1945 F. I. Frankl established a generalization of these problems for the well-known Chaplygin equation

subject to a certain Frankl condition. In 1953 and 1955 M. H. Protter generalized these problems

even further by improving the Frankl condition. In 1977 we generalized these results in several n-

dimensional simply connected domains. In 1990 we proposed the exterior Tricomi problem in a doubly

connected domain. In 2002 we considered uniqueness of quasi-regular solutions for a bi-parabolic

elliptic bi-hyperbolic Tricomi problem. In 2006 G. C. Wen investigated the exterior Tricomi problem

for general mixed type equations. In this paper we establish uniqueness of quasi-regular solutions

for the exterior Tricomi and Frankl problems for quaterelliptic - quaterhyperbolic mixed type partial

differential equations of second order with eight parabolic degenerate lines and propose certain open

problems. These mixed type boundary value problems are very important in fluid mechanics.

2000 Mathematics Subject Classifications: 35MO5.

Key Words and Phrases: Quasi-regular solution, Tricomi equation, Chaplygin equation, quaterelliptic

equation, quaterhyperbolic equation, Tricomi problem.

1. Introduction

In 1904 S. A. Chaplygin [11] pointed out that the nonlinear equation of an adiabatic

potential perfect gas:

(ρ2α2 −ψy
2)ψx x + 2ψxψyψx y + (ρ

2α2 −ψx
2)ψy y = 0,

is closely connected with the study of the linear mixed type equation

K(y)ux x + uy y = 0

Email addresses: jrassias�primedu.uoa.gr; jrass�otenet.gr
http://www.ejpam.com 186 c© 2011 EJPAM All rights reserved.



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 187

named Chaplygin equation, where ψ = ψ(x , y) is the stream function, α := local velocity of

sound and ρ := density of gas.

In 1923 F. G. Tricomi [19] initiated the work on boundary value problems for linear partial

differential mixed type equations of second order and related equations of variable type. The

well-known mixed type partial differential equation was called Tricomi equation:

yux x + uy y = 0

after F. G. 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.

In 1945 F. I. Frankl [3] drew attention to the fact that the Tricomi problem was closely con-

nected to the study of gas flow with nearly sonic speeds. In 1953 and 1955 M. H. Protter [7]

generalized and improved the afore-mentioned results in the euclidean plane. In 1977 we

[8] generalized these results in Rn (n > 2). In 1982 we [9] established a maximum principle

of the Cauchy problem for hyperbolic equations in Rn+1 (n ≥ 2). In 1983 we [10] solved the

Tricomi problem with two parabolic lines of degeneracy and, in 1992, we [12] established

the well-posedness of the Tricomi problem in euclidean regions. Interesting results for the

Tricomi problem were achieved by G. Baranchev [1] in 1986, and M. Kracht and E. Kreyszig

[4] in 1986, as well. Related information was reported by G. Fichera [2] in 1985, and E.

Kreyszig [5-6] in 1989 and 1994. Our [11,14-15] work, in 1990 and 1999, was in analogous

areas of mixed type equations. In 1990-2009, G. C. Wen et al. [17,20-28] have applied the

complex analytic method and achieved fundamental uniqueness and existence results for solu-

tions of the Tricomi and Frankl problems for classical mixed type partial differential equations

with boundary conditions. In 1993 R.I. Semerdjieva [18] introduced the hyperbolic equation

K1(y)ux x + (K2(y)uy)y + ru = f in the lower half-plane. In 1997 we [13] considered the

more general case of the above hyperbolic equation, so that it was elliptic in the upper half-

plane and parabolic on the line y = 0. In 2002, we [16] considered the more general Tricomi

problem with partial differential equation the new bi-parabolic elliptic bi-hyperbolic equation

Lu ≡ K1(y)(M2(x)ux)x +M1(x)(K2(y)uy)y + r(x , y)u= f (x , y), (1)

which is parabolic on both segments x = 0, 0 < y ≤ 1; y = 0, 0 < x ≤ 1, elliptic in the

Euclidean region Ge = {(x , y) ∈ G(⊂ R2) : x > 0, y > 0} and hyperbolic in both regions

Gh1
= {(x , y) ∈ G(⊂ R2) : x > 0, y < 0} ; Gh2

= {(x , y) ∈ G(⊂ R2) : x < 0, y > 0}, with G

the mixed domain of (1). In 1999 we [15] proved existence of weak solutions for a particular

Tricomi problem. Then we established uniqueness of quasi-regular solutions [8,10-13,16] for

the Tricomi problem. However, the question about the uniqueness of quasi-regular solutions

and the existence of weak solutions for the Tricomi and Frankl problems associated to the said

mixed type equation (1) for even more general doubly connected mixed domain is still open.

In particular via this paper we propose and investigate the exterior Tricomi and Frankl prob-

lems for quaterelliptic and quaterhyperbolic equations with eight parabolic lines of degener-

acy and establish uniqueness of quasi-regular solutions. Also we propose new open problems.



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 188

These results are interesting in Aerodynamics and Hydrodynamics. 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.

Definition 1. The Tricomi problem or Problem T consists of 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 σ (for y > 0) with

endpoints O = (0,0) and A= (1,0) and by two real hyperbolic characteristics Γ, γ of the Tricomi

equation satisfying the pertinent characteristic equation such that these characteristics Γ, γ meet

at a point P (for y < 0) with Γ emanating from A and γ from O,

Γ : x +
2

3
(−y)3/2 = 1 and γ : x −

2

3
(−y)3/2 = 0

and u assumes prescribed continuous boundary values on both arcs σ and γ. 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 hyperolic region; and the segment OA is parabolic.

Definition 2. A function u = u(x , y) is a regular solution of Problem T in the sense of F. G.

Tricomi if:

1) u is continuous in the closure of D which is the union of D with its boundary consisting of

the three curves σ,Γ,γ;

2) The first order partial derivatives of u are continuous in the closure of D except 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 both arcs σ,γ.

2. The Exterior Tricomi Problem

Consider the quaterelliptic - quaterhyperbolic equation (1) with eight parabolic lines of

degeneracy in a bounded doubly connected mixed domain D with a piecewise smooth bound-

ary ∂ D, where f = f (x , y) is continuous in D, r = r(x , y) is once-continuously differen-

tiable in D, Ki = Ki(y) (i = 1,2) are once-continuously differentiable for y ∈ [−k1, k2] with

−k1 = in f {y : (x , y) ∈ D} and k2 = sup{y : (x , y) ∈ D}, and Mi = Mi(x) (i = 1,2) are

once-continuously differentiable for x ∈ [−m1, m2] with −m1 = in f {x : (x , y) ∈ D} and

m2 = sup{x : (x , y) ∈ D}.



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 189

Figure 1

Besides,

K1(y)







> 0 for {y < 0} ∪ {y > 1}
= 0 for {y = 0} ∪ {y = 1} ;

< 0 for {0< y < 1}

M1(x)







> 0 for {x < −1} ∪ {x > 0}
= 0 for {x = 0} ∪ {x = −1},
< 0 for {−1< x < 0}

as well as K2 = K2(y)> 0, M2 = M2(x)> 0, everywhere in D, so that

K = K(y) =
K1(y)

K2(y)







> 0 for {y < 0} ∪ {y > 1}
= 0 for {y = 0} ∪ {y = 1} ;

< 0 for {0< y < 1}



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 190

M = M(x) =
M1(x)

M2(x)







> 0 for {x < −1} ∪ {x > 0}
= 0 for {x = 0} ∪ {x = −1}.
< 0 for {−1< x < 0}

The boundary ∂ D = E x t(D) ∪ Int(D) of the doubly connected domain D is formed by the

following two exterior and interior boundaries

E x t(D) = (Γ0 ∪Γ0
′ ∪Γ0

′′ ∪Γ0
′′′)∪ (Γ2 ∪Γ2

′)∪ (γ2 ∪ γ2
′)∪ (∆1 ∪∆1

′)∪ (δ1 ∪δ1
′),

Int(D) = (Γ1 ∪Γ1
′)∪ (γ1 ∪ γ1

′)∪ (∆2 ∪∆2
′)∪ (δ2 ∪δ2

′),

respectively: In the right hyperbolic domain G2 = {(x , y) ∈ D : 0 < x < 1, 0 < y < 1} with

boundary ∂ G2 = (O1B1)∪(O2B2)∪(Γ1∪Γ1
′)∪(Γ2∪Γ2

′), where O1B1, O2B2 are two parabolic

lines with end points O1 = (0,1), B1 = (1,1) and O2 = (0,0), B2 = (1,0) and Γ1,Γ1
′,Γ2,Γ2

′

are four characteristics, so that:

Γ1 :

∫ x

0

p

M(t)d t = −

∫ y

1

p

−K(t)d t : 0< x < 1,
1

2
< y < 1 emanating from O1 = (0,1),

Γ1
′ :

∫ x

0

p

M(t)d t =

∫ y

0

p

−K(t)d t : 0< x < 1 , 0< y <
1

2
, emanating from O2 = (0,0),

Γ2 :

∫ x

1

p

M(t)d t =

∫ y

1

p

−K(t)d t : 0< x < 1 ,
1

2
< y < 1,emanating from B1 = (1,1),

Γ2
′ :

∫ x

1

p

M(t)d t = −

∫ y

0

p

−K(t)d t : 0< x < 1,0< y <
1

2
,emanating from B2 = (1,0),

where M = M(x) > 0, 0 < x < 1 and K = K(y) < 0, 0 < y < 1. In the upper hyperbolic

domain G2
′ = {(x , y) ∈ D : −1< x < 0,1< y < 2} with boundary

∂ G2
′ = (O1Z1) ∪ (O1

′E1) ∪ (γ1 ∪ γ1
′) ∪ (γ2 ∪ γ2

′), where O1Z1, O1
′E1 are two parabolic lines

with end points O1 = (0,1), Z1 = (0,2) and O1
′ = (−1,1), E1 = (−1,2) and γ1,γ1

′,γ2,γ2
′ are

four characteristics, so that:

γ1 :

∫ x

0

p

−M(t)d t = −

∫ y

1

p

K(t)d t : −
1

2
< x < 0 , 1< y < 2, emanating from O1 = (0,1),

γ1
′ :

∫ x

−1

p

−M(t)d t =

∫ y

1

p

K(t)d t : −1< x < −
1

2
, 1< y < 2, emanating from O1

′ = (−1,1),

γ2 :

∫ x

0

p

−M(t)d t =

∫ y

2

p

K(t)d t : −
1

2
< x < 0, 1< y < 2, emanating from Z1 = (0,2),

γ2
′ :

∫ x

−1

p

−M(t)d t = −

∫ y

2

p

K(t)d t :−1< x < −
1

2
,1< y < 2, emanating from E1 = (−1,2),



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 191

where M = M(x) < 0, −1 < x < 0 and K = K(y) > 0, 1 < y < 2. In the left hyperbolic

domain G2
′′ = {(x , y) ∈ D : −2< x < −1,0< y < 1} with boundary

∂ G2
′′ = (O1

′A1) ∪ (O2
′A2) ∪ (∆1 ∪∆1

′) ∪ (∆2 ∪∆2
′), where O1

′A1, O2
′A2 are two parabolic

lines with end points O1
′ = (−1,1), A1 = (−2,1) and O2

′ = (−1,0), A2 = (−2,0) and

∆1,∆1
′,∆2,∆2

′ are four characteristics, so that:

∆1 :

∫ x

−2

p

M(t)d t = −

∫ y

1

p

−K(t)d t :−2< x < −1,
1

2
< y < 1, emanating from A1 = (−2,1),

∆1
′ :

∫ x

−2

p

M(t)d t =

∫ y

0

p

−K(t)d t : −2< x < −1, 0< y <
1

2
, emanating from A2 = (−2,0),

∆2 :

∫ x

−1

p

M(t)d t =

∫ y

1

p

−K(t)d t : −2< x < −1,
1

2
< y < 1, emanating from O1

′ = (−1,1),

∆2
′ :

∫ x

−1

p

M(t)d t = −

∫ y

0

p

−K(t)d t :−2< x < −1,0< y <
1

2
, emanating from O2

′ = (−1,0),

where M = M(x) > 0, −2 < x < −1 and K = K(y) < 0, 0 < y < 1. In the lower hyperbolic

domain G2
′′′ = {(x , y) ∈ D : −1< x < 0,−1< y < 0} with boundary

∂ G2
′′′ = (O2Z2)∪ (O2

′E2)∪ (δ1 ∪δ1
′)∪ (δ2 ∪δ2

′), where O2Z2, O2
′E2 are two parabolic lines

with end points O2 = (0,0), Z2 = (0,−1) and O2
′ = (−1,0), E2 = (−1,−1) and δ1,δ1

′,δ2,δ2
′

are four characteristics, so that:

δ1 :

∫ x

0

p

−M(t)d t = −

∫ y

−1

p

K(t)d t : −
1

2
< x < 0, −1< y < 0, emanating from Z2 = (0,−1),

δ1
′ :

∫ x

−1

p

−M(t)d t =

∫ y

−1

p

K(t)d t : −1< x < −
1

2
,−1< y < 0, emanating from E2 = (−1,−1),

δ2 :

∫ x

0

p

−M(t)d t =

∫ y

0

p

K(t)d t : −
1

2
< x < 0, −1< y < 0, emanating from O2 = (0,0),

δ2
′ :

∫ x

−1

p

−M(t)d t = −

∫ y

0

p

K(t)d t :−1< x < −
1

2
,−1< y < 0, starting from O2

′ = (−1,0),

where M = M(x) < 0, −1 < x < 0 and K = K(y) > 0, 1 < y < 2. In the upper right elliptic

domain G1 = {(x , y) ∈ D : x > 0, y > 1} with boundary ∂ G1 = (O1B1)∪ (O1Z1)∪ Γ0, where

O1B1, O1Z1 are two parabolic lines with end points O1 = (0,1), B1 = (1,1) and

O1 = (0,1), Z1 = (0,2) and Γ0 is the upper right elliptic arc connecting points B1 = (1,1)

and Z1 = (0,2). In the lower right elliptic domain G1
′ = {(x , y) ∈ D : x > 0, y < 0} with

boundary ∂ G1
′ = (O2B2) ∪ (O2Z2) ∪ Γ0

′, where O2B2, O2Z2 are two parabolic lines with end

points O2 = (0,0), B2 = (1,0) and O2 = (0,0), Z2 = (0,−1) and Γ0
′ is the lower right el-

liptic arc connecting points B2 = (1,0) and Z2 = (0,−1). In the upper left elliptic domain

G1
′′ = {(x , y) ∈ D : x < −1, y > 1} with boundary ∂ G1

′′ = (O1
′E1) ∪ (O1

′A1) ∪ Γ0
′′,



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 192

where O1
′E1, O1

′A1 are two parabolic lines with end points O1
′ = (−1,1), E1 = (−1,2) and

O1
′ = (−1,1), A1 = (−2,1) and Γ0

′′ is the upper left elliptic arc connecting points A1 = (−2,1)

and E1 = (−1,2). In the lower left elliptic domain G1
′′′ = {(x , y) ∈ D : x < −1, y < 0} with

boundary ∂ G1
′′′ = (O2

′A2) ∪ (O2
′E2) ∪ Γ0

′′′, where O2
′E2, O2

′A2 are two parabolic lines with

end points O2
′ = (−1,0), E2 = (−1,−1) and O2

′ = (−1,0),A2 = (−2,0) and Γ0
′′′ is the lower

left elliptic arc connecting points A2 = (−2,0) and E2 = (−1,−1).

Let us consider the intersection points of the hyperbolic characteristics:

Γ1 ∩ Γ1
′ = {P1}, where P1 = (x1, 1

2
), 0 < x1 < 1; Γ2 ∩ Γ2

′ = {P2}, where P2 = (x2, 1

2
),

0< x1 <
1

2
< x2 < 1;∆1∩∆1

′ = {P1
′}, where P1

′ = (x1
′, 1

2
), −2< x1

′ < −1;∆2∩∆2
′ = {P2

′},

where P2
′ = (x2

′, 1

2
), −2 < x1

′ < −3

2
< x2

′ < −1; γ1 ∩ γ1
′ = {Q1}, where Q1 = (−

1

2
, y1),

1 < y1 < 2; γ2 ∩ γ2
′ = {Q2}, where Q2 = (−

1

2
, y2), 1 < y1 <

3

2
< y2 < 2; δ1 ∩ δ1

′ = {Q1
′},

where Q1
′ = (−1

2
, y1
′), −1< y1

′ < 0; δ2 ∩δ2
′ = {Q2

′}, where Q2
′ = (−1

2
, y2
′),

−1< y1
′ < −1

2
< y2

′ < 0. If we denote Θ = Θ(x) =
p

|M(x)|, H = H(y) =
p

|K(y)|, we set

D1(x) =

∫ x

0

Θ(t)d t, D2(x) =

∫ x

1

Θ(t)d t,

D3(x) =

∫ x

−1

Θ(t)d t, D4(x) =

∫ x

−2

Θ(t)d t,

G1(y) =

∫ y

0

H(t)d t, G2(y) =

∫ y

1

H(t)d t,

G3(y) =

∫ y

−1

H(t)d t, G4(y) =

∫ y

2

H(t)d t.

Domains G1, G2 differ in notation from functions G1(y), G2(y). Thus, we have the following

equations for the hyperbolic characteristics

Γ1 ∪ γ1 : D1(x) = −G2(y), Γ1
′ ∪ γ1

′ : G2(y) =

¨

D1(x) on Γ1
′

D3(x) on γ1
′ ,

Γ2 ∪ γ2 :

¨

D2(x) = G2(y) on Γ2

D1(x) = G4(y) on γ2
, Γ2
′ ∪ γ2

′ :

¨

D2(x) = −G1(y) on Γ2
′

D3(x) = −G4(y) on γ2
′

∆1 ∪δ1 :

¨

D4(x) = −G2(y) on ∆1

D1(x) = −G3(y) on δ1
, ∆1

′ ∪δ1
′ :

¨

D4(x) = G1(y) on ∆1
′

D3(x) = G3(y) on δ1
′

∆2 ∪δ2 :

¨

D3(x) = G2(y) on ∆2

D1(x) = G1(y) on δ2
, ∆2

′ ∪δ2
′ : D3(x) = |G1(y)|.

Note that:



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 193

1) The boundary ∂ D is assumed to be a piecewise continuously differentiable arc. The

elliptic arcs are “star-shaped” (counterclockwise).

2) We consider continuous solutions u of the quaterelliptic-quaterhyperbolic equation (1)

with eight parabolic lines, which have the property that ux , uy are continuous in the clo-

sure D̄ = D ∪ ∂ D. These continuity conditions may be weakened at the following eight

points A1,A2, B1, B2,O1,O2,O1
′,O2
′, by considering ux , uy continuous on the bound-

ary ∂ D except at these points. By “quaterelliptic” and “quaterhyperbolic” we mean

that equation (1) is elliptic in four different subdomains and hyperbolic in four other

subdomains of the whole domain D. In fact, equation (1) is elliptic and hyperbolic in

G1 ∪ G1
′ ∪ G1

′′ ∪ G1
′′′ and G2 ∪ G2

′ ∪ G2
′′ ∪ G2

′′′, respectively.

Definition 3. A function u= u(x , y) is a quasi-regular solution [7,8,10-16] of Problem (ET) if

i) u ∈ C2(D)∩ C(D), D = D ∪ ∂ D;

ii) the Green’s theorem (of the integral calculus) is applicable to the integrals

∫∫

D

ux Lud xd y ,

∫∫

D

uy Lud xd y;

iii) the boundary and region integrals, which arise, exist; and

iv) u satisfies the mixed type equation (1) in D and the following boundary condition on the

exterior boundary E x t(D) :

u=























ϕ1(s) on Γ0; ϕ2(s) on Γ0
′

ϕ3(s) on Γ0
′′; ϕ4(s) on Γ0

′′′

ψ1(x) on Γ2; ψ2(x) on Γ2
′

ψ3(x) on γ2; ψ4(x) on γ2
′

ψ5(x) on ∆1; ψ6(x) on ∆1
′

ψ7(x) on δ1; ψ8(x) on δ1
′

(2)

with continuous prescribed values.

The Exterior Tricomi Problem or Problem (ET): consists of finding a solution u of the

quaterelliptic -quaterhyperbolic equation (1) with eight parabolic lines in D and which as-

sumes continuous prescribed values (2).

Uniqueness Theorem 1. Consider the quaterelliptic - quaterhyperbolic equation (1) with eight

parabolic lines and the boundary condition (2). Assume the above mixed doubly connected do-

main D and the following conditions:

(R1) r ≤ 0 on the interior boundary Int(D),



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 194

(R2)







xd y − (y − 1)d x ≥ 0 on Γ0

xd y − yd x ≥ 0 on Γ0
′

(x + 1)d y − (y − 1)d x ≥ 0 on Γ0
′′

(x + 1)d y − yd x ≥ 0 on Γ0
′′′

,

(R3)







































2r + x rx + (y − 1)ry ≤ 0 in G1

2r + x rx + yry ≤ 0 in G1
′

2r + (x + 1)rx + (y − 1)ry ≤ 0 in G1
′′

2r + (x + 1)rx + yry ≤ 0 in G1
′′′

r + x rx ≤ 0 in G2

r + (y − 1)ry ≤ 0 in G2
′

r + (x + 1)rx ≤ 0 in G2
′′

r + yry ≤ 0 in G2
′′′

,

(R4) Ki > 0, Mi > 0 (i = 1,2), in G1 ∪ G1
′ ∪ G1

′′ ∪ G1
′′′,

(R5)

¨

K1 < 0 , M1 > 0 in G2 ∪ G2
′′

K1 > 0 , M1 < 0 in G2
′ ∪ G2

′′′ ,

(R6)







Ṁ1 ≥ 0, Ṁ2 ≥ 0 ; K1
′ ≥ 0, K2

′ ≥ 0 in G1

Ṁ1 ≥ 0, Ṁ2 ≥ 0 ; K1
′ ≤ 0, K2

′ ≤ 0 in G1
′

Ṁ1 ≤ 0, Ṁ2 ≤ 0 ; K1
′ ≥ 0, K2

′ ≥ 0 in G1
′′

Ṁ1 ≤ 0, Ṁ2 ≤ 0 ; K1
′ ≤ 0, K2

′ ≤ 0 in G1
′′′

,

(R7) K2 > 0, M2 > 0 in D,

(R8)







Ṁ1 ≥ 0, Ṁ2 ≤ 0 in G2

K1
′ ≥ 0, K2

′ ≤ 0 in G2
′

Ṁ1 ≤ 0, Ṁ2 ≥ 0 in G2
′′

K1
′ ≤ 0, K2

′ ≥ 0 in G2
′′′

.

Let ()x = ∂ ()/∂ x , ()· = d()/d x , ()y = ∂ ()/∂ y, ()′ = d()/d y, where f = f (x , y) is

continuous in D, r = r(x , y) is once-continuously differentiable in D, Ki = Ki(y) (i = 1,2)

are once-continuously differentiable for y ∈ [−k1, k2] with −k1 = in f {y : (x , y) ∈ D} and

k2 = sup{y : (x , y) ∈ D}, and Mi = Mi(x) (i = 1,2) are once-continuously differentiable for

x ∈ [−m1, m2] with −m1 = in f {x : (x , y) ∈ D} and m2 = sup{x : (x , y) ∈ D}. Then the

Problem (ET) has at most one quasi-regular solution in D.

Proof. We apply the well-known a-b-c energy integral method with a = 0, and use the

above mixed type equation (1) as well as the boundary condition (2). First, we assume two

quasi-regular solutions u1,u2 of the Problem (ET). Then we claim that u = u1 − u2 = 0 holds

in the domain D. In fact, we investigate

0 = J = 2< lu, Lu >0=

∫∫

D

2luLud xd y (3)



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 195

where lu = b(x)ux + c(y)uy , and Lu= L(u1−u2) = Lu1− Lu2 = f − f = 0 in D, with choices

b = b(x) =







x inG1 ∪ G1
′ ∪ G2

x + 1 inG1
′′ ∪ G1

′′′ ∪ G2
′′

0 inG2
′ ∪ G2

′′′
,

c = c(y) =







y in G1
′ ∪ G1

′′′ ∪ G2
′′′

y − 1 in G1 ∪ G1
′′ ∪ G2

′

0 in G2 ∪ G2
′′

. (4)

We consider the new differential identities

2bK1M2ux ux x =
�

bK1M2ux
2
�

x − (bM2)x K1ux
2,

2bK2M1ux uy y =
�

2bK2M1uxuy

�

y − 2bM1K2
′ux uy −
�

bK2M1uy
2
�

x + (bM1)x K2uy
2,

2cK1M2uyux x =
�

2cK1M2ux uy

�

x − 2cK1Ṁ2ux uy −
�

cK1M2ux
2
�

y + (cK1)
′M2ux

2,

2cK2M1uyuy y =
�

cK2M1uy
2
�

y − (cK2)
′M1uy

2,

2bruux = (bru2)x − (br)xu2, 2cruuy = (cru2)y − (cr)yu2,

as well as t1 is the coefficient of ux in Lu, or

t1 = t1(x , y) = K1(y)Ṁ2(x), (5)

and t2 is the coefficient of uy in Lu, or

t2 = t2(x , y) = K2
′(y)M1(x). (6)

Employing these identities and the classical Green’s theorem of the integral calculus we obtain

from (1),(3),(5), and (6) that

0 = J =

∫∫

D

2(bux + cuy )
�

K1(M2ux)x +M1(K2uy)y + ru
�

d xd y

=

∫∫

D

2(bux + cuy )
�

K1M2ux x + K2M1uy y + t1ux + t2uy + ru
�

d xd y,

= ID + I∂ D, (7)

where ID =
∫∫

D

Q(ux ,uy)d xd y =
∫∫

D

(Aux
2 + Buy

2 +Γu2 + 2∆uxuy)d xd y;

I∂ D =

∫

∂ D

Q̃(ux ,uy )ds =

∫

∂ D

(Ãux
2 + B̃uy

2+ Γ̃u2 + 2∆̃uxuy)ds,



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 196

with A= −K1(bM2)x + (cK1)
′M2 + 2bt1, B = K2(bM1)x − (cK2)

′M1 + 2ct2,

Γ = −
�

(br)x + (cr)y
�

,

∆= −
�

bK2
′M1 + cK1Ṁ2 − bt2 − ct1

�

= −
�

b(K2
′M1 − t2) + c(K1Ṁ2 − t1)

�

= 0

(because of (5) and (6)) in D, and Ã= (bv1 − cv2)K1M2, B̃ = (−bv1+ cv2)K2M1,

Γ̃ = (bv1 + cv2)r, ∆̃ = bK2M1v2 + cK1M2v1 on ∂ D, where v = (v1, v2) = (d y/ds,−d x/ds) is

the outer unit normal vector on the boundary ∂ D of the domain D such that

ds2 = d x2+ d y2 > 0, |v|= 1 ;
∫∫

D

()x d xd y =
∫

∂ D

()v1ds,
∫∫

D

()y d xd y =
∫

∂ D

()v2ds are the Green’s integral formulas. From

the above conditions, we obtain

0≤ A =







xK1Ṁ2 + (y − 1)K1
′M2 in G1

xK1Ṁ2 + yK1
′M2 in G1

′

(x + 1)K1Ṁ2 + (y − 1)K1
′M2 in G1

′′

(x + 1)K1Ṁ2 + yK1
′M2 in G1

′′′

;

0≤ A =







−K1(M2 − x Ṁ2) in G2

M2

�

K1 + (y − 1)K1
′� in G2

′

−K1

�

M2 − (x + 1)Ṁ2

�

in G2
′′

M2

�

K1 + yK1
′� in G2

′′′

.

Similarly we get

0≤ B =







xK2Ṁ1 + (y − 1)K2
′M1 in G1

xK2Ṁ1 + yK2
′M1 in G1

′

(x + 1)K2Ṁ1 + (y − 1)K2
′M1 in G1

′′

(x + 1)K2Ṁ1 + yK2
′M1 in G1

′′′

;

0≤ B =







K2(M1 + x Ṁ1) in G2

−M1

�

K2− (y − 1)K2
′� in G2

′

K2

�

M1 + (x + 1)Ṁ1

�

in G2
′′

−M1

�

K2− yK2
′� in G2

′′′

.

Also

0≤ Γ = −







2r + x rx + (y − 1)ry in G1

2r + x rx + yry in G1
′

2r + (x + 1)rx + (y − 1)ry in G1
′′

2r + (x + 1)rx + yry in G1
′′′

;

0≤ Γ = −







r + x rx in G2

r + (y − 1)ry in G2
′

r + (x + 1)rx in G2
′′

r + yry in G2
′′′

.



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 197

Therefore,

ID =

∫ ∫

D

=

∫ ∫

G1∪G1
′∪G1

′′∪G1
′′′

+

∫ ∫

G2∪G2
′∪G2

′′∪G2
′′′

=
�

∫∫

G1

+

∫∫

G1
′

+

∫∫

G1
′′

+

∫∫

G1
′′′

�

+
�

∫∫

G2

+

∫∫

G2
′

+

∫∫

G2
′′

+

∫∫

G2
′′′

�

≥ 0.

We claim that I∂ D = IE x t(D)
⋃

I nt(D) = IE x t(D) + II nt(D) ≥ 0, where

IE x t(D) =
�

∫

Γ0

+

∫

Γ0
′

+

∫

Γ0
′′

+

∫

Γ0
′′′

�

+
�

∫

Γ2

+

∫

Γ2
′

�

+
�

∫

γ2

+

∫

γ2
′

�

+
�

∫

∆1

+

∫

∆1
′

�

+
�

∫

δ1

+

∫

δ1
′

�

≥ 0,

and

II nt(D) =
�

∫

Γ1

+

∫

Γ1
′

�

+
�

∫

γ1

+

∫

γ1
′

�

+
�

∫

∆2

+

∫

∆2
′

�

+
�

∫

δ2

+

∫

δ2
′

�

≥ 0.

In fact, on Γ0 with b = x , c = y + 1 :

Ã=
�

x v1− (y + 1)v2

�

K1M2 ≥ 0 , B̃ =
�

− x v1+ (y + 1)v2

�

K2M1 ≥ 0;

Γ̃ =
�

x v1+ (y + 1)v2

�

r ≥ 0 ; ∆̃ = xK2M1v2 + (y + 1)K1M2v1.

IΓ0
=

∫

Γ0

Q̃(ux ,uy )ds+

∫

Γ0

Γ̃u2ds

=

∫

Γ0

N2
�

x v1+ (y + 1)v2

�

Hds+

∫

Γ0

�

x v1+ (y + 1)v2

�

ru2ds,

=

∫

Γ0

N2
�

xd y − (y + 1)d x
�

H ≥ 0

where u|Γ0
= 0 and

0= du= ux d x + uy d y = N
�

v1d x + v2d y
�

; ux = N v1, uy = N v2,

with a normalizing factor N , and H|Γ0
= K1M2v1

2+K2M1v2
2 > 0, as well as Γ0 is a “star-liked”

arc, such that xd y − (y + 1)d x |Γ0
≥ 0 and

Q̃ = Q̃(ux ,uy ) = [Ãux
2 + B̃uy

2 + 2∆̃uxuy]|Γ0
= N2
�

x v1+ (y + 1)v2

�

H ≥ 0.

Similarly, we obtain

0 ≤ IΓ0∪Γ0
′∪Γ0

′′∪Γ0
′′′ =

∫

Γ0

N2
�

xd y − (y + 1)d x
�

H +

∫

Γ0
′

N2
�

xd y − yd x
�

H



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 198

+

∫

Γ0
′′

N2
�

(x + 1)d y − (y − 1)d x
�

H +

∫

Γ0
′′′

N2
�

(x + 1)d y − yd x
�

H.

Also on Γ2 ∪Γ2
′ with b = x , c = 0 :

IΓ2∪Γ2
′ =

∫

Γ2∪Γ2
′

Q̃(ux ,uy )ds+

∫

Γ2∪Γ2
′

Γ̃u2ds

=

∫

Γ2∪Γ2
′

N2(x v1)Hds+

∫

Γ2∪Γ2
′

(x v1)ru2ds = 0,

where u|Γ2∪Γ2
′ = 0 and H|Γ2∪Γ2

′ = K1M2v1
2 + K2M1v2

2 = 0, because both Γ2, Γ2
′ are charac-

teristics.

Similarly we get

Iγ2∪γ2
′ =

∫

γ2∪γ2
′

N2
�

(y − 1)v2

�

Hds+

∫

γ2∪γ2
′

�

(y − 1)v2

�

ru2ds = 0,

I∆1∪∆1
′ =

∫

∆1∪∆1
′

N2
�

(x + 1)v1

�

Hds+

∫

∆1∪∆1
′

�

(x + 1)v1

�

ru2ds = 0,

Iδ1∪δ1
′ =

∫

δ1∪δ1
′
N2
�

yv2

�

Hds+

∫

δ1∪δ1
′

�

yv2

�

ru2ds = 0.

Also on Γ1 ∪Γ1
′ with b = x , c = 0 :

Ã= (x v1)K1M2 ≥ 0, B̃ = (−x v1)K2M1 ≥ 0;

Γ̃ = (x v1)r ≥ 0, ∆̃ = xK2M1v2.

IΓ1∪Γ1
′ =

∫

Γ1∪Γ1
′

Q̃(ux ,uy)ds+

∫

Γ1∪Γ1
′

Γ̃u2ds

=

∫

Γ1∪Γ1
′

�

x
�

(K1M2v1)ux
2 + (−K2M1v1)uy

2 + 2(K2M1v2)uxuy

�	

,

+

∫

Γ1∪Γ1
′

(x v1)ru2ds > 0

because r|Γ1∪Γ1
′ ≤ 0, v1|Γ1∪Γ1

′ < 0, and H = 0 since both Γ1, Γ1
′ are characteristics as well as

ÃB̃− (∆̃)2 = −x2K2M1H = 0.

Similarly, we get

Iγ1∪γ1
′ =

∫

γ1∪γ1
′

Q̃(ux ,uy)ds+

∫

γ1∪γ1
′

Γ̃u2ds



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 199

=

∫

γ1∪γ1
′

�

(y − 1)
�

(−K1M2v2)ux
2 + (K2M1v2)uy

2 + 2(K1M2v1)uxuy

�	

,

+

∫

γ1∪γ1
′

�

(y − 1)v2

�

ru2ds > 0

I∆2∪∆2
′ =

∫

∆2∪∆2
′

Q̃(ux ,uy )ds+

∫

∆2∪∆2
′

Γ̃u2ds

=

∫

∆2∪∆2
′

�

(x + 1)
�

(K1M2v1)ux
2 + (−K2M1v1)uy

2 + 2(K2M1v2)uxuy

�	

,

+

∫

∆2∪∆2
′

�

(x + 1)v1

�

ru2ds > 0

Iδ2∪δ2
′ =

∫

δ2∪δ2
′
Q̃(ux ,uy)ds+

∫

δ2∪δ2
′
Γ̃u2ds

=

∫

δ2∪δ2
′

�

y
�

(−K1M2v2)ux
2 + (K2M1v2)uy

2 + 2(K1M2v1)uxuy

�	

+

∫

δ2∪δ2
′

�

yv2

�

ru2ds > 0.

From (7) we get 0= ID + I∂ D ≥ 0 with ID ≥ 0 and I∂ D ≥ 0. These relations yield

ID = I∂ D = 0. (8)

It is clear that
∫∫

Ḡ1

Γu2d xd y = −
∫∫

Ḡ1

�

2r + x rx + (y − 1)ry

�

u2d xd y = 0.

Therefore, we get u(x , y) = 0 everywhere in Ḡ1. Alternatively

∫∫

Ḡ1

��

xK1Ṁ2 + (y − 1)K1
′M2

�

ux
2 +
�

xK2Ṁ1 + (y − 1)K2
′M1

�

uy
2
�

d xd y = 0,

yielding ux ≡ 0; uy ≡ 0 in Ḡ1. Thus, in Ḡ1 : u(x , y)≡ 0.

Similarly

u(x , y)≡ 0 in Ḡ1 ∪ Ḡ′1 ∪ Ḡ′′1 ∪ Ḡ′′′1 . (9)

It is clear that
∫∫

Ḡ2

Γu2d xd y = −
∫∫

Ḡ2

(r + x rx)u
2d xd y = 0.

Therefore, we get u(x , y)≡ 0 everywhere in Ḡ2. Alternatively

∫∫

Ḡ2

��

xK1Ṁ2

�

ux
2 +
�

xK2Ṁ1

�

uy
2
�

d xd y = 0,



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 200

yielding ux ≡ 0; uy ≡ 0 in Ḡ2. Thus in Ḡ2 : u(x , y)≡ 0.

Similarly

u(x , y)≡ 0 in Ḡ2 ∪ Ḡ′2 ∪ Ḡ′′2 ∪ Ḡ′′′2 .

From (8) we get

I∂ D = IE x t(D) + II nt(D) = IΓ0∪Γ0
′∪Γ0

′′∪Γ0
′′′

+
�

IΓ1∪Γ1
′ + Iγ1∪γ1

′ + I∆2∪∆2
′ + Iδ2∪δ2

′
�

= 0.

Thus,

IΓ1∪Γ1
′ = 0. (10)

Also alternatively, by a well-known theorem on hyperbolic equations if u|Γ2∪Γ2
′ = 0 (from the

boundary condition) and IΓ1∪Γ1
′ = 0 (from (10)), then u(x , y) ≡ 0 everywhere in Ḡ2.

Pertinent to the above, there is the following general uniqueness approach:

First, from the maximum principle, if u|
G2

⋃

G
′
2

⋃

G
′′
2

⋃

G
′′′
2

= 0, it follows that

u|
G1

⋃

G
′
1

⋃

G
′′
1

⋃

G
′′′
1

= 0. Second, from the uniqueness of the solution of the Cauchy problem,

if u|
G1

⋃

G
′
1

⋃

G
′′
1

⋃

G
′′′
1

= 0, it follows u|
G2

⋃

G
′
2

⋃

G
′′
2

⋃

G
′′′
2

= 0. Thus, u(x , y) ≡ 0 everywhere in D,

completing the proof of the uniqueness theorem.

Note that the case: r = r(x , y) = 0 in D and K1
′(0) = Ṁi(0) = 0 (i = 1,2) yield also

uniqueness results for the problem (ET).

3. The Exterior Frankl Problem

Consider the quaterelliptic-quaterhyperbolic equation (1) with eight parabolic lines of de-

generacy in a bounded doubly connected mixed domain D̃ with a piecewise smooth boundary

∂ D̃ = E x t(D̃)∪ Int(D̃) =
�

E x tEl(D̃)∪ E x tHn(D̃)
�

∪ Int(D̃),

where D̃ is a part of D, and Int(D̃) = Int(D), as well as

E x tEl(D̃) = Γ0 ∪Γ0
′ ∪Γ0

′′ ∪Γ0
′′′

is the elliptic exterior boundary of D̃ and

E x tHn(D̃) = (Γ̃2 ∪ Γ̃
′
2)∪ (γ̃2 ∪ γ̃

′
2)∪ (∆̃1 ∪ ∆̃

′
1)∪ (δ̃1 ∪ δ̃

′
1)

is the non-characteristic hyperbolic exterior boundary of D̃, such that:

E x t(D̃) = E x tEl(D̃)∪ E x tHn(D̃) =
�

Γ0 ∪ Γ0
′ ∪Γ0

′′ ∪ Γ0
′′′�

∪
�

(Γ̃2 ∪ Γ̃
′
2)∪ (γ̃2 ∪ γ̃

′
2)∪ (∆̃1 ∪ ∆̃

′
1)∪ (δ̃1 ∪ δ̃

′
1)
�

is the exterior boundary of D̃, with the following non-characteristics:

Γ̃2, Γ̃′2, γ̃2, γ̃′2, ∆̃1, ∆̃′1, δ̃1, δ̃′1 :



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 201

Figure 2

Γ̃2 :
p

M(x)d x ≥
p

−K(y)d y ;

∆̃′1 : 0≤
p

M(x)d x ≤
p

−K(y)d y ;

Γ̃′2 :
p

M(x)d x ≤ −
p

−K(y)d y ≤ 0 ;

∆̃1 :
p

M(x)d x ≥ −
p

−K(−y)d y ;

γ̃2 : 0≥
p

−M(x)d x ≥
p

K(y)d y ;

δ̃′1 : 0≤
p

−M(x)d x ≤
p

K(y)d y ;

γ̃′2 : 0≥
p

−M(x)d x ≥ −
p

K(y)d y ;

δ̃1 : 0≤
p

−M(x)d x ≤ −
p

K(y)d y ,

or

Γ̃2 ∪ ∆̃
′
1 : 0≤

d y

d x
≤

p

M(x)
p

−K(y)
; Γ̃′2 ∪ ∆̃1 : 0≥

d y

d x
≥ −

p

M(x)
p

−K(y)
;



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 202

γ̃2 ∪ δ̃
′
1 :

d y

d x
≥

p

−M(x)
p

K(y)
; γ̃′2 ∪ δ̃1 :

d y

d x
≤ −

p

−M(x)
p

K(y)
,

satisfying the non-characteristic relation

H = K1M2v1
2 + K2M1v2

2 ≥ 0, or K(y)(d y)2 +M(x)(d x)2 ≥ 0,

and intersecting characteristics Γ1, Γ′1, γ1, γ′1, ∆2, ∆′2, δ2, δ′2, only once.

Note that

(Γ̃2 ∪ ∆̃1)∪ (Γ̃
′
2 ∪ ∆̃

′
1) : (
p

M(x)d x −
p

−K(y)d y)(
p

M(x)d x +
p

−K(y)d y) = H;

(γ̃2 ∪ δ̃1)∪ (γ̃
′
2 ∪ δ̃

′
1) : (
p

−M(x)d x −
p

K(y)d y)(
p

−M(x)d x +
p

K(y)d y) = −H.

Let us consider the intersection points of the hyperbolic characteristics:

Γ̃2 ∩ Γ̃
′
2 = {P̃2}, where P̃2 = ( x̃2, 1

2
), 0< x1 <

1

2
< x̃2 < x2 < 1; ∆̃1 ∩ ∆̃

′
1 = {P̃

′
1}, where

P̃ ′1 = ( x̃
′
1, 1

2
), −2< x1 < x̃ ′1 <−1; γ̃2 ∩ γ̃

′
2 = {Q̃2}, where Q̃2 = (−

1

2
, ỹ2),

1< y1 <
3

2
< ỹ2 < y2 < 2; δ̃1 ∩ δ̃

′
1 = {Q̃

′
1}, where Q̃′1 = (−

1

2
, ỹ ′1), −1< y1

′ < ỹ ′1 < 0.

Let the right hyperbolic domain G̃2 ⊂ G2 = {(x , y) ∈ D : 0< x < 1,0< y < 1} with boundary

∂ G̃2 = (O1B1)∪ (O2B2)∪ (Γ1 ∪Γ1
′)∪ (Γ̃2 ∪ Γ̃

′
2).

Let the upper hyperbolic domain G̃′2 ⊂ G2
′ = {(x , y) ∈ D : −1 < x < 0,1 < y < 2} with

boundary ∂ G̃′2 = (O1Z1)∪ (O1
′E1)∪ (γ1 ∪ γ1

′)∪ (γ̃2 ∪ γ̃
′
2).

Let the left hyperbolic domain G̃′′2 ⊂ G2
′′ = {(x , y) ∈ D : −2 < x < −1,0 < y < 1} with

boundary ∂ G̃′′2 = (O1
′A1)∪ (O2

′A2)∪ (∆̃1 ∪ ∆̃
′
1)∪ (∆2 ∪∆2

′).

Let the lower hyperbolic domain G̃′′′2 ⊂ G2
′′′ = {(x , y) ∈ D : −1 < x < 0,−1 < y < 0} with

boundary ∂ G̃′′′2 = (O2Z2)∪ (O2
′E2)∪ (δ̃1 ∪ δ̃

′
1)∪ (δ2 ∪δ2

′).

Assume boundary conditions on the above exterior boundary E x t(D̃) :

u =























ϕ1(s) on Γ0; ϕ2(s) on Γ0
′

ϕ3(s) on Γ0
′′; ϕ4(s) on Γ0

′′′

ψ̃1(x) on Γ̃2; ψ̃2(x) on Γ̃′2
ψ̃3(x) on γ̃2; ψ̃4(x) on γ̃′2
ψ̃5(x) on ∆̃1; ψ̃6(x) on ∆̃′1
ψ̃7(x) on δ̃1; ψ̃8(x) on δ̃′1

(11)

with continuous prescribed values.

The Exterior Frankl Problem or Problem (EF): consists of finding a solution u of the

quaterelliptic -quaterhyperbolic equation (1) with eight parabolic lines in D̃(⊂ D) and which

assumes continuous prescribed values (11).

Uniqueness Theorem 2. Consider the quaterelliptic - quaterhyperbolic equation (1) with eight

parabolic lines and the boundary condition (11). Assume the above mixed doubly connected

domain D̃(⊂ D) and the following conditions:



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 203

(R1) r ≤ 0 on the interior boundary Int(D̃)(= Int(D)),

(R2)







xd y − (y − 1)d x ≥ 0 on Γ0

xd y − yd x ≥ 0 on Γ0
′

(x + 1)d y − (y − 1)d x ≥ 0 on Γ0
′′

(x + 1)d y − yd x ≥ 0 on Γ0
′′′

,

(R3)







































2r + x rx + (y − 1)ry ≤ 0 in G1

2r + x rx + yry ≤ 0 in G1
′

2r + (x + 1)rx + (y − 1)ry ≤ 0 in G1
′′

2r + (x + 1)rx + yry ≤ 0 in G1
′′′

r + x rx ≤ 0 in G̃2

r + (y − 1)ry ≤ 0 in G̃′2
r + (x + 1)rx ≤ 0 in G̃′′2
r + yry ≤ 0 in G̃′′′2

,

(R4) Ki > 0, Mi > 0 (i = 1,2), in G1 ∪ G1
′ ∪ G1

′′ ∪ G1
′′′,

(R5)

¨

K1 < 0 , M1 > 0 in G̃2 ∪ G̃′′2
K1 > 0 , M1 < 0 in G̃′2 ∪ G̃′′′2

,

(R6)







Ṁ1 ≥ 0, Ṁ2 ≥ 0 ; K1
′ ≥ 0, K2

′ ≥ 0 in G1

Ṁ1 ≥ 0, Ṁ2 ≥ 0 ; K1
′ ≤ 0, K2

′ ≤ 0 in G1
′

Ṁ1 ≤ 0, Ṁ2 ≤ 0 ; K1
′ ≥ 0, K2

′ ≥ 0 in G1
′′

Ṁ1 ≤ 0, Ṁ2 ≤ 0 ; K1
′ ≤ 0, K2

′ ≤ 0 in G1
′′′

,

(R7) K2 > 0, M2 > 0 in D̃(⊂ D),

(R8)







Ṁ1 ≥ 0, Ṁ2 ≤ 0 in G̃2

K1
′ ≥ 0, K2

′ ≤ 0 in G̃′2
Ṁ1 ≤ 0, Ṁ2 ≥ 0 in G̃′′2
K1
′ ≤ 0, K2

′ ≥ 0 in G̃′′′2

.

Let ()x = ∂ ()/∂ x , ()· = d()/d x , ()y = ∂ ()/∂ y, ()′ = d()/d y, where f = f (x , y) is

continuous in D̃(⊂ D), r = r(x , y) is once-continuously differentiable in D̃(⊂ D),

Ki = Ki(y) (i = 1,2) are once-continuously differentiable for y ∈ [−k1, k2] with

−k1 = in f {y : (x , y) ∈ D̃(⊂ D)} and k2 = sup{y : (x , y) ∈ D̃(⊂ D)}, and Mi = Mi(x)

(i = 1,2) are once-continuously differentiable for x ∈ [−m1, m2] with

−m1 = in f {x : (x , y) ∈ D̃(⊂ D)} and m2 = sup{x : (x , y) ∈ D̃(⊂ D)}. Then the Problem (EF)

has at most one quasi-regular solution in D̃(⊂ D).

Proof. We apply the well-known energy integral method, and use the above mixed type

equation (1) as well as the boundary condition (11). First, we assume two quasi-regular

solutions u1,u2 of the Problem (EF). Then we claim that u= u1 − u2 = 0 holds in the domain



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 204

D̃(⊂ D). In fact, we investigate

0 = J̃ = 2< l̃u, Lu >0=

∫∫

D̃

2l̃uLud xd y

where l̃u = b̃(x)ux + c̃(y)uy , and Lu = L(u1 − u2) = Lu1 − Lu2 = f − f = 0 in D̃(⊂ D) with

choices

b̃ = b̃(x) =







x in G1 ∪ G1
′ ∪ G̃2

x + 1 in G1
′′ ∪ G1

′′′ ∪ G̃′′2
0 in G̃′2 ∪ G̃′′′2

,

and

c̃ = c̃(y) =







y in G1
′ ∪ G1

′′′ ∪ G̃′′′2

y − 1 in G1 ∪ G1
′′ ∪ G̃′2

0 in G̃2 ∪ G̃′′2

.

The rest of the proof is similar to the proof of the uniqueness theorem 1 (for the exterior

Tricomi problem), except clearly proving in additional that the following condition holds on

the non-characteristic hyperbolic exterior boundary

E x tHn(D̃) = (Γ̃2 ∪ Γ̃
′
2)∪ (γ̃2 ∪ γ̃

′
2)∪ (∆̃1 ∪ ∆̃

′
1)∪ (δ̃1 ∪ δ̃

′
1) :

0< b̃v1+ c̃ v2 =











x v1 on Γ̃2 ∪ Γ̃
′
2

(y − 1)v2 on γ̃2 ∪ γ̃
′
2

(x + 1)v1 on ∆̃1 ∪ ∆̃
′
1

yv2 on δ̃1 ∪ δ̃
′
1

.

Uniqueness Theorem 3. Consider the quaterelliptic - quaterhyperbolic equation (1) with eight

parabolic lines and the boundary condition (11). Assume the above mixed doubly connected

domain D̃(⊂ D) and the following conditions:

(R1) r ≤ 0 on the interior boundary Int(D̃)(= Int(D)),

(R2)







xd y − (y − 1)d x ≥ 0 on Γ0

xd y − yd x ≥ 0 on Γ0
′

(x + 1)d y − (y − 1)d x ≥ 0 on Γ0
′′

(x + 1)d y − yd x ≥ 0 on Γ0
′′′

,

(R3)







































2r + x rx + (y − 1)ry ≤ 0 in G1

2r + x rx + yry ≤ 0 in G1
′

2r + (x + 1)rx + (y − 1)ry ≤ 0 in G1
′′

2r + (x + 1)rx + yry ≤ 0 in G1
′′′

r + x rx ≤ 0 in G̃2

r + (y − 1)ry ≤ 0 in G̃′2
r + (x + 1)rx ≤ 0 in G̃′′2
r + yry ≤ 0 in G̃′′′2

,



J. Rassias / Eur. J. Pure Appl. Math, 4 (2011), 186-208 205

(R4) Ki > 0, Mi > 0 (i = 1,2), in G1 ∪ G1
′ ∪ G1

′′ ∪ G1
′′′,

(R5)

¨

K1 < 0 , M1 > 0 in G̃2 ∪ G̃′′2
K1 > 0 , M1 < 0 in G̃′2 ∪ G̃′′′2

,

(R6) b(x)Ki(y)Ṁ j(x)+ c(y)Ki
′(y)M j(x)> 0, (1≤ i 6= j ≤ 2) in G1∪G1

′∪G1
′′∪G1

′′′, where

b(x) =

¨

x in G1 ∪ G1
′

x + 1 in G1
′′ ∪ G1

′′′ ;

c(y) =

¨

y − 1 in G1 ∪ G1
′′

y in G1
′ ∪ G1

′′′ .

(R7) K2 > 0, M2 > 0 in D̃(⊂ D),

(R8)
¨

Mi(x)− (−1)i b̃(x)Ṁi(x)> 0 in G̃2 ∪ G̃′′2
Ki(y)− (−1)i c̃(x)Ki

′(y)> 0 in G̃′2 ∪ G̃′′′2

(i ∈ {1,2}),

where

b̃(x) =







x in G̃2

x + 1 in G̃′′2
0 in G̃′2 ∪ G̃′′′2

;

c̃(y) =







y − 1 in G̃′2
y in G̃′′′2

0 in G̃2 ∪ G̃′′2

.

Let us denote ()x = ∂ ()/∂ x , ()· = d()/d x , ()y = ∂ ()/∂ y, ()′ = d()/d y, where

f = f (x , y) is continuous in D̃(⊂ D), r = r(x , y) is once-continuously differentiable in

D̃(⊂ D), Ki = Ki(y) (i = 1,2) are once-continuously differentiable for y ∈ [−k1, k2] with

−k1 = in f {y : (x , y) ∈ D̃(⊂ D)} and k2 = sup{y : (x , y) ∈ D̃(⊂ D)}, and Mi = Mi(x)

(i = 1,2) are once-continuously differentiable for x ∈ [−m1, m2] with

−m1 = in f {x : (x , y) ∈ D̃(⊂ D)} and m2 = sup{x : (x , y) ∈ D̃(⊂ D)}. Then the Problem (EF)

has at most one quasi-regular solution in D̃(⊂ D).

4. Open Problems

4.1. Extend “quasi-regularity” of solutions to “regularity” by fixing singularities at the fol-

lowing twelve points:

O1 = (0,1), O1
′ = (−1,1), O2 = (0,0),O2

′ = (−1,0);

A1 = (−2,1), B1 = (1,1), A2 = (−2,0), B2 = (1,0);

E1 = (−1,2), Z1 = (0,2), E2 = (−1,−1), Z2 = (0,−1).



REFERENCES 206

4.2. Investigate the exterior Tricomi and Frankl problems in a multiply connected mixed

domain.

4.3. Establish “well-posedness” of solutions for the exterior Tricomi and Frankl problems, in

the sense that there is at most one quasi-regular solution and a weak solution exists.

4.4. Solve the n - dimensional Tricomi and Frankl problems in a multiply connected mixed

domain.

4.5. Establish the extremum principle for the exterior Tricomi problem:

“A solution of the exterior Tricomi (or Frankl) problem, vanishing on the exterior bound-

ary of the considered mixed domain, achieves neither a positive maximum nor a negative

minimum on open arcs of the type-degeneracy curves.”

4.6. Solve the Tricomi problem for PDE of second order:

4.6.1 K(y − xm− xn)ux x + uy y + r(x , y)u= f (x , y);

4.6.2 ux x +M(x − ym− yn)uy y + r(x , y)u= f (x , y);

4.6.3 K(xm+ yn − 1)ux x + uy y + r(x , y)u= f (x , y), for example m= n= 2 or = 2/3;

4.6.4 K
�

(y − xm)(y − xn)
�

ux x + uy y + r(x , y)u = f (x , y);

4.6.5 K(y − xn)ux x +M(x − ym)uy y + r(x , y)u= f (x , y);

4.6.6 K(yk − xm± xn)ux x +M(x k − ym± yn)uy y + r(x , y)u= f (x , y);

4.6.7 K
�

ym(y − xn)
�

ux x +M
�

xm(x − yn)
�

uy y + r(x , y)u = f (x , y);

4.6.8 K
�

(y − xm)(y − xn)
�

ux x +M
�

(x − yα)(x − yβ )
�

uy y + r(x , y)u= f (x , y).

4.7. Solve the Tricomi problem for PDE of fourth order:

�

sgn(y − x l)|y − x l |k
∂ 2

∂ x2
+ sgn(x − yn)|x − yn|m

∂ 2

∂ y2
+ r
�2

u = f .

4.8. Solve the 3 - dimensional Tricomi problem for mixed type PDE of second order:

sgn(z)|z|k(ux x ± uy y ) + sgn(x y)|x |m|y|nuzz + ru= f .

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