Abstract
Full Text
MATHEMATICS
M. S. SALAKHITDINOV
ON SOME WELL-POSED PROBLEMS FOR THE EQUATION
[
y^m u_{xxx}+u_{xyy}=0
]
(Presented by Academician I. N. Vekua on 17 XII 1962)
1. In the present communication we shall discuss certain boundary-value problems for the equation of mixed-composite type
[
y^m u_{xxx}+u_{xyy}=0,
\tag{1}
]
where (m) is an odd positive integer.
Denote by (D_1) a finite simply connected domain bounded by the segment (A(-1,0)B(1,0)) of the (x)-axis and by a simple smooth Jordan arc (\sigma), lying in the half-plane (y>0) and resting on the (x)-axis at the points (A) and (B). Suppose that each straight line (y=c), (0\le c\le h), intersects the arc (\sigma) at two points, the straight line (y=h) has the single common point (C_1) (a point of tangency) with the arc (\sigma), and the straight lines (y=c), (c>h), have no common points with the arc (\sigma).
Let (D_2) denote the domain bounded by the segment (AB) of the (x)-axis and by the two characteristics
[
AC:\ x=\frac{2}{m+2}(-y)^{(m+2)/2}-1
]
and
[
BC:\ x=-\frac{2}{m+2}(-y)^{(m+2)/2}+1
]
of equation (1), issuing from the points
[
C\left[0,\ -\left(\frac{m+2}{2}\right)^{2/(m+2)}\right].
]
By (D) we shall denote the union of the domains (D_1) and (D_2) together with the open segment (AB) of the (x)-axis.
Problem A. Find a solution (u(x,y)) of equation (1), regular in the domain (D_1), continuous up to the contour of this domain, and satisfying the boundary conditions:
[
u=f \text{ on } \sigma,\qquad
u=f_1 \text{ on } AB,\qquad
u_n=\varphi \text{ on } \sigma_1,
\tag{2}
]
where (\sigma_1) is the part (AC_1) of the arc (\sigma); (n) is the interior normal to (\sigma_1); (f), (f_1), (\varphi) are prescribed continuous functions.
Problem A*. Find a solution (u(x,y)) of equation (1), regular in the domain (D_1), continuous up to the contour of this domain, and satisfying the conditions:
[
u=f \text{ on } \sigma,\qquad
u_y=f_2 \text{ on } AB,\qquad
u_n=\varphi \text{ on } \sigma_1,
\tag{3}
]
where (f), (f_2), (\varphi) are prescribed continuous functions.
Problem B. It is required to find a solution (u(x,y)) of equation (1), regular in the domain (D_2), continuous in the closed domain (\overline{D}_2), and satisfying the conditions:
[
u=f_1 \text{ on } AB,\qquad
u=\psi,\qquad
u_n=\varphi_1 \text{ on } AC,
\tag{4}
]
where the prescribed functions (f_1,\varphi_1\in C^{(2)}), and (\psi\in C^{(3)}).
Problem B*. Find a solution (u(x,y)) of equation (1), regular in the domain (D_2), continuous in the closed domain (\overline{D}_2), and satisfying the conditions:
[
u_y=f_2 \text{ on } AB,\qquad
u=\psi,\qquad
u_n=\varphi_1 \text{ on } AC,
\tag{5}
]
where (f_2,\varphi_1\in C^{(2)}), (\psi\in C^{(3)}).
Problem C (mixed problem). It is required to determine in the domain (D) a solution (u(x,y)) of equation (1), continuous up to the contour and satisfying the boundary conditions:
[
u=f \text{ on } \sigma,\qquad u_n=\varphi \text{ on } \sigma_1,\qquad
u=\psi,\qquad u_n=\varphi_1 \text{ on } AC,
\tag{6}
]
where the given functions (f,\varphi\in C,\ \varphi_1\in C^{(2)},\ \psi\in C^{(3)}).
We shall assume that (A) and (B) are not corner points of the domain (D).
These problems were posed in communication (1). As is known, any regular solution of equation (1) can be represented in the form (3)
[
u(x,y)=z(x,y)+\omega(y),
\tag{7}
]
where (z(x,y)) is a solution of the equation
[
y^m z_{xx}+z_{yy}=0,
\tag{8}
]
and (\omega(y)) is an arbitrary twice continuously differentiable function. In considering the problems formulated above we shall assume that everywhere along the arc (\sigma_1) (except, of course, for its end (C_1)) the condition
[
\cos(n,x)\ne 0
\tag{9}
]
holds.
2. In studying problem A, without loss of generality one may assume that in the representation (7) of the solutions of equation (1) the arbitrary function (\omega(y)) is subject to the conditions
[
\omega(0)=\omega(C_1)=0.
\tag{10}
]
The uniqueness of the solution of this problem under conditions (9) and (10) was proved in (1).
We shall prove the existence of a solution of problem A in the case when (\sigma) coincides with the so-called normal contour:
[
x=\cos\theta,\qquad
y=\left(\frac{m+2}{2}\sin\theta\right)^{2/(m+2)}.
]
Without loss of generality, it may be assumed that the functions (f,f_1) vanish at the points (A) and (B). In addition, we shall assume that the functions (f'(\theta), f_1(x)) and (\varphi(\theta)) satisfy a Hölder condition.
On the basis of (2) and (7), for the solution (z(x,y)) of equation (8) we have the boundary conditions
[
z=f-\omega \text{ on } \sigma,\qquad
z=f_1 \text{ on } AB,\qquad
z_n=\varphi-\omega'\cos(n,y) \text{ on } \sigma_1.
\tag{11}
]
A solution of equation (8), regular in the domain (D_1) and satisfying the first two of conditions (11), is expressed by formula (5):
[
\begin{aligned}
z(R,\theta_0)
&=k\left(\frac{m+2}{2}R\sin\theta_0\right)^{\frac{2}{m+2}}
\int_{-1}^{1} f_1(\xi)
\left[
\left(R^2-2R\xi\cos\theta_0+\xi^2\right)^{\beta-1}
\right.\
&\qquad\left.
-\left(1-2R\xi\cos\theta_0+R^2\xi^2\right)^{\beta-1}
\right]\,d\xi\
&\quad+\frac{k(m+4)}{2}
\left(\frac{m+2}{2}R\sin\theta_0\right)^{\frac{2}{m+2}}
(1-R^2)
\int_0^\pi
\left{
f(\theta)
-\omega\left[\left(\frac{m+2}{2}\sin\theta\right)^{2/(m+2)}\right]
\right}\
&\qquad\times
(r_1^2)^{\beta-2}
F\left(1-\beta,\,2-\beta,\,2-2\beta,\,1-\frac{r^2}{r_1^2}\right)
\sin\theta\,d\theta,
\tag{12}
\end{aligned}
]
[
\frac{r^2}{r_1^2}=1+R^2-2R\cos(\theta\mp\theta_0),\qquad
R^2=x^2+\frac{4}{(m+2)^2}y^{m+2},
]
[
\beta=\frac{m}{2m+4},\qquad
k=\frac{1}{4\pi}\left(\frac{4}{m+2}\right)^{2-2\beta}
\frac{\Gamma^2(1-\beta)}{\Gamma(2-2\beta)}.
]
Realizing the third of conditions (11), for determining the unknown function (\omega(y)) from (12) we obtain the singular integral equation
[
\sin \theta_0 \,\gamma(\theta_0)+\frac{1}{\pi}\int_{\pi/2}^{\pi}
\frac{K(\theta_0,\theta)\,\gamma(\theta)}{\theta-\theta_0}\,d\theta
=\delta(\theta_0),
\tag{13}
]
where
[
\gamma(\theta_0)=\omega'\left[\left(\frac{m+2}{2}\sin\theta_0\right)^{2/(m+2)}\right],
]
and (\delta(\theta_0)) is expressed in terms of the prescribed functions. Equation (13) is an equation of normal type, and its index is equal to unity.
By virtue of the equivalence of problem A and equation (13), on the basis of (10) we conclude that the homogeneous equation corresponding to equation (13) has one nontrivial solution, which is constant. Hence, on the basis of the general theory of singular integral equations for open contours ((^4)), the solvability of equation (13) follows at once. This proves the existence of a solution of problem A.
- Let us consider problem (A^*). In the representation (7) of solutions of equation (1), without loss of generality one may assume that
[
\omega(C_1)=\omega'(0)=0.
\tag{14}
]
Consequently, the homogeneous problem (A^*) is reduced to determining in the domain (D_1) a regular solution (z(x,y)) of equation (8) satisfying the conditions:
[
z=-\omega \text{ on } \sigma,\qquad
z_y=0 \text{ on } AB,\qquad
z_n=-\omega'\cos(n,y) \text{ on } \sigma_1.
\tag{15}
]
A regular solution (z(x,y)) of equation (8) inside the domain (D_1) does not attain extremal values. On the basis of (9), (14), and (15), the function (z(x,y)) cannot attain extremal values also on the contour of the domain (D_1). Hence it follows that the homogeneous problem (A^) cannot have a solution different from zero. The existence of a solution of problem (A^) is readily proved under the same assumptions that were imposed on (\sigma) and on the prescribed functions in proving the existence of a solution of problem A.
- In considering problems B and (B^*), without loss of generality we subject the arbitrary function (\omega(y)) to the conditions
[
\omega(0)=\omega'(0)=0.
\tag{16}
]
The solution of these problems can be written explicitly on the basis of Darboux’s formula. Let us write the solution of problem (B^). Obviously, without loss of generality, we may assume that (f_2(A)=\psi'(A)=\varphi_1(A)=0). Then the solution of problem (B^) is represented in the form:
[
\begin{aligned}
u(x,y)=&\;2^{2/(m+2)}\gamma_1
\int_{-1}^{1}
\left{
\Phi\left[x+\frac{2}{m+2}(-y)^{(m+2)/2}t\right]\right.\
&\left.\qquad\qquad
+ f_2^*\left[x+\frac{2}{m+2}(-y)^{(m+2)/2}t\right]
\right}
(1-t^2)^{\beta-1}\,dt\
&+\left(\frac{2}{m+2}\right)^{2/(m+2)}
\gamma_2 y\int_{-1}^{1}
f_2\leftx+\frac{2}{m+2}(-y)^{(m+2)/2}t\right^{-\beta}\,dt\
&-\gamma_1\int_0^1
\Phi\left[-1+\frac{4}{m+2}(-y)^{(m+2)/2}t\right]
t^{\beta-1}(1-t)^{\beta-1}\,dt\
&+\psi\left[\frac{2}{m+2}(-y)^{(m+2)/2}-1\right],
\end{aligned}
]
[
\gamma_1=\Gamma(2\beta)/\Gamma^2(\beta),\qquad
\gamma_2=\left(\frac{m+2}{4}\right)^{2/(m+2)}
\Gamma(2-2\beta)/\Gamma^2(1-\beta),
]
where the function (\Phi) is determined through (\psi) and (\varphi_1), and (f_2^) through (f_2). The uniqueness of the solution of problems B and (B^) is obvious.
- We now consider problem C. The function (\omega(y)=\omega_2(y)), (-\left(\dfrac{m+2}{2}\right)^{2/(m+2)} \leqslant y \leqslant 0), entering into representation (7), is determined uniquely by the last two conditions from (6). Consequently, the solution of problem C is reduced to finding a solution (z(x,y)), regular in the domain (D), of equation (8), satisfying the conditions:
[
z=f-\omega_1 \text{ on } \sigma,\qquad
z=\psi-\omega_2 \text{ on } AC,\qquad
z_n=\varphi-\omega_1'\cos(n,y) \text{ on } \sigma_1,
\tag{17}
]
where (\omega_1(y)=\omega(y)), (0\leqslant y\leqslant h). Without loss of generality we shall again assume that condition (14) holds. Thus the homogeneous problem C has been reduced to determining a solution (z(x,y)), regular in the domain (D), of equation (8), satisfying the conditions:
[
z=-\omega_1 \text{ on } \sigma,\qquad
z=0 \text{ on } AC,\qquad
z_n=-\omega'_n\cos(n,y) \text{ on } \sigma_1.
]
Hence, on the basis of the well-known extremum principle for the Tricomi problem ({}^{2}) and conditions (9), (14), we conclude that the homogeneous problem C cannot have a nonzero solution. The existence of a solution of problem C is not difficult to prove in the case when (\sigma) is a normal contour, the function (f) is twice continuously differentiable, and (\varphi(\theta)) satisfies the Hölder condition. Let us first note that the first two conditions in (17) constitute a Tricomi problem for equation (8). Solving this problem by the known method ({}^{2}), we determine the function (z_y(x,0)=\nu(x)), which will be expressed in terms of the unknown function (\omega_1(y)) and the given functions (f,\psi,\varphi_1). After this, satisfying the third of conditions (17), in order to determine the unknown function (\omega_1(y)) we obtain a singular integral equation of normal type of the form (13), the solvability of which is easy to show on the basis of the known theory of singular integral equations.
I take this opportunity to express my sincere gratitude to my teacher A. V. Bitsadze for valuable advice and guidance.
Institute of Mathematics named after V. I. Romanovskii
Academy of Sciences of the Uzbek SSR
Received
14 XII 1962
REFERENCES
- A. V. Bitsadze, “Some problems of mathematics and mechanics” (for the sixtieth anniversary of Acad. M. A. Lavrent’ev), Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk, 1961, p. 47.
- A. V. Bitsadze, Equations of mixed type, Publishing House of the Academy of Sciences of the USSR, 1959.
- A. V. Bitsadze, M. S. Salakhitdinov, Siberian Mathematical Journal, 2, No. 1, 79 (1961).
- N. I. Muskhelishvili, Singular integral equations, Moscow, 1962.
- S. Gellerstedt, Ark. Mat. Astr. och. Fys., 25 A, No. 10, 1 (1935).