UDC 517.944

ON THE THEORY OF BOUNDARY VALUE PROBLEMS FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATIONS WITH TYPE DEGENERATION ON THE BOUNDARY OF THE DOMAIN

D. K. GVAZAVA

Let \(D\) be a domain lying in the upper half-plane \(y>0\), bounded by the contour \(\Gamma=AB+\sigma\), where \(AB\) is a segment of the axis \(y=0\), and \(\sigma\) is a simple smooth curve for which the following assumptions are made: 1) it approaches the line \(y=0\) orthogonally at the points \(A(-1,0)\) and \(B(1,0)\); 2) the functions \(x=x(s)\), \(y=y(s)\), which give a parametric representation of this curve, have second derivatives satisfying the Hölder condition; \(s\) is the length of the arc \(\Gamma\), measured from a fixed point in the positive direction.

Consider the equation

\[ E[u(x,y)] \equiv y^m u_{xx}+u_{yy}=f(x,y,u), \tag{1} \]

where \(m\) is a given positive number, and the given single-valued function \(f(x,y,u)\) has continuous first derivatives for all \((x,y)\in \overline D\) and for all finite values of \(u(x,y)\), with

\[ f'_u(x,y,u)\geq 0. \]

Equation (1) is elliptic for \(y>0\), while for \(y=0\) it degenerates parabolically.

Let \(F(s)\) be a continuous function given on the boundary \(\Gamma\) of the domain \(D\).

Theorem 1. There exists a unique solution \(u(x,y)\), twice continuously differentiable in the domain \(D\), of equation (1), continuous in the closed domain \(D\) and satisfying the boundary condition

\[ u(x,y)\big|_\Gamma=F(s). \tag{2} \]

Since \(f'_u(x,y,u)\) is nonnegative for all \((x,y)\in \overline D\) and for all finite \(u\), the uniqueness of the solution of problem (1)—(2) follows from the extremum principle (see [3]).

We shall prove the existence of a solution by the method proposed by L. Lichtenstein in [8].

Introduce into equation (1) the parameter \(\lambda\) as follows:

\[ E[u(x,y)]=\lambda f(x,y,u),\quad 0\leq \lambda\leq 1. \tag{3} \]

For \(\lambda=1\), equation (3) coincides with equation (1), while for \(\lambda=0\) we obtain the linear homogeneous equation

\[ E[u(x,y)]=0. \tag{4} \]

Let us now note that for every solution of the linear differential equation

\[ y^m u_{xx}+u_{yy}+c(x,y)u=f(x,y),\quad c(x,y)\leq 0, \]

satisfying the boundary condition (2), the estimate holds

\[ |u(x,y)| \leq k\left(\max_{\Gamma}|F(s)|+\max_{\overline D}|f(x,y)|\right), \tag{5} \]

\[ k=\mathrm{const},\qquad (x,y)\in \overline D . \]

The validity of estimate (5) is proved by the method indicated in [2].

We write equation (3) in the form

\[ E[u(x,y)]-\lambda f'_u(x,y,\tilde u)u=\lambda f(x,y,0), \]

where \(\tilde u\) lies between zero and \(u\). Applying inequality (5) to this equation, we obtain

\[ |u(x,y)|\leq k\left(\max_{\Gamma}|F(s)|+\max_{\overline D}|f(x,y,0)|\right). \]

Suppose that in the three-dimensional domain

\[ P\{(x,y)\in D,\ |u|<3K\}, \]

where

\[ \mathrm{const}=K>k\left(\max_{\Gamma}|F(s)|+\max_{\overline D}|f(x,y,0)|\right), \]

the function \(f(x,y,u)\) has a derivative \(f''_{uu}(x,y,u)\) satisfying a Lipschitz condition with respect to the last argument,

\[ |f''_{uu}(x,y,u_1)-f''_{uu}(x,y,u_2)|\leq L|u_1-u_2|,\quad L=\mathrm{const}, \]

and continuous in the closed domain \(\overline P\).

The solution \(u_0(x,y)\) of equation (4), satisfying the boundary condition (2), exists uniquely (see [1, 6]) and, in the case of the so-called normal domain, when the curve \(\sigma\) is given by the equation

\[ x^2+\frac{4}{(m+2)^2}y^{m+2}=1, \]

is written explicitly with the aid of the Green’s function

\[ G(x,y;x_1,y_1)= \frac{yy_1}{(m+2)^{\frac{m+4}{m+2}}} \left\{ E\left[x,\frac{y^{m+2}}{(m+2)^2};x_1,\frac{y_1^{m+2}}{(m+2)^2}\right] -\right. \]

\[ \left. -\rho^{-\frac{m+4}{m+2}} E\left[x,\frac{y^{m+2}}{(m+2)^2};\frac{x_1}{\rho^2}; \frac{y_1^{m+2}}{\rho^4(m+2)^2}\right] \right\}, \]

where

\[ \rho^2=x^2+\frac{4}{(m+2)^2}y^{m+2} \]

and

\[ E(x,y;\xi,\eta)=\frac{2^{\frac{2}{m+2}}}{\pi} \int_0^\pi \left[(x-\xi)^2+4y+4\eta-\right. \]

\[ \left. -8\sqrt{y\eta}\cos\beta \right]^{-\frac{m+4}{2m+4}} (\sin\beta)^{\frac{2}{m+2}}\,d\beta \]

(see [5, 9]).

On the basis of the properties of the Green’s function, as in potential theory, problem (3)—(2), for any value of the parameter \(\lambda>0\), reduces to the equivalent integral equation

\[ u(x,y)=-\lambda \iint_D G(x,y;x_1,y_1) f[x_1,y_1;u(x_1,y_1)]\,dx_1dy_1+u_0(x,y). \tag{6} \]

Every solution \(u_\lambda(x,y)\) of the nonlinear integral equation (6) is bounded for any \(0\leq \lambda\leq 1\).

Indeed, since the function \(f(x,y,u)\) is continuous in the closed domain \(\overline P\), there exists a positive constant \(M\) such that \(|f(x,y,u)|\leq M\).

In view of the fact that \(G(x,y;x_1,y_1)>0\) everywhere in the domain \(D\), from the integral equation (6) we obtain

\[ u_\lambda(x,y)\leq u_0(x,y)+\lambda \iint_D G(x,y;x_1,y_1)|f(x_1,y_1;u_\lambda)|\,dx_1dy_1\leq \]

\[ \leq u_0(x,y)+M\lambda \iint_D G(x,y;x_1,y_1)\,dx_1dy_1\leq u_0(x,y)+Mg, \]

where

\[ g=\max_{\overline D}\iint_D G(x,y;x_1,y_1)\,dx_1dy_1. \]

Analogously we obtain \(u_\lambda(x,y)\geq u_0(x,y)-Mg\).

Thus,

\[ u_0(x,y)-Mg\leq u_\lambda(x,y)\leq u_0(x,y)+Mg. \tag{7} \]

Suppose that for \(\lambda=\nu<1\) a solution \(u_\nu(x,y)\) of problem (3)—(2) is known:

\[ E[u_\nu(x,y)]=\nu f(x,y,u_\nu),\qquad u_\nu(x,y)\big|_{\Gamma}=F(s). \tag{8} \]

We shall prove the existence of a solution \(u_{\nu+\alpha}(x,y)\) of the analogous problem for \(\lambda=\nu+\alpha\),

\[ E[u_{\nu+\alpha}(x,y)]=(\nu+\alpha)f(x,y,u_{\nu+\alpha}),\qquad u_{\nu+\alpha}\big|_{\Gamma}=F(s) \tag{9} \]

for some \(\alpha>0\).

Denoting by \(\xi(x,y)\) the difference \(u_{\nu+\alpha}(x,y)-u_\nu(x,y)\) and subtracting identity (8) from (9), we obtain

\[ E[\xi(x,y)]=(\nu+\alpha)f(x,y,u_{\nu+\alpha})-\nu f(x,y,u_\nu)= \]

\[ =(\nu+\alpha)\bigl[f(x,y,u_{\nu+\alpha})-f(x,y,u_\nu)\bigr] +\alpha f(x,y,u_\nu), \]

or

\[ E[\xi(x,y)] =\alpha f(x,y,u_\nu)+(\nu+\alpha) \left[ f'_u(x,y,u_\nu)\xi(x,y)+ \frac{1}{2}f''_{uu}(x,y,u_\nu+\vartheta\xi)\xi^2(x,y) \right], \quad 0<\vartheta<1. \tag{10} \]

It is obvious that the function \(\xi(x,y)\) satisfies the boundary condition

\[ \xi(x,y)\big|_{\Gamma}=0. \tag{11} \]

In view of (6), (10), and (11), we have

\[ \xi(x,y)=-\iint_D G(x,y;x_1,y_1)\left\{\alpha f[x_1,y_1;u_\nu(x_1,y_1)]+\right. \]

\[ +(\nu+\alpha)f'_u[x_1,y_1;u_\nu(x_1,y_1)]\xi(x_1,y_1) +\frac{\nu+\alpha}{2}f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+ \vartheta\xi(x_1,y_1)]\xi^2(x_1,y_1)\}dx_1dy_1, \]

or, what is the same,

\[ \begin{aligned} \xi(x,y)&+(\nu+\alpha)\iint_D G(x,y;x_1,y_1) f'_u[x_1,y_1;u_\nu(x_1,y_1)]\xi(x_1,y_1)\,dx_1dy_1\\ &=-\alpha\iint_D G(x,y;x_1,y_1)f[x_1,y_1;u_\nu(x_1,y_1)]\,dx_1dy_1\\ &\quad-\frac{1}{2}(\nu+\alpha)\iint_D G(x,y;x_1,y_1) \{f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\\ &\qquad+\vartheta\xi(x_1,y_1)]\xi^2(x_1,y_1)\}\,dx_1dy_1 . \end{aligned} \tag{12} \]

The integral equation (12) is equivalent to the problem (10)—(11). From the inequalities (7) we have

\[ d=\max_{\overline D}|\xi(x,y)|\le Mg . \tag{13} \]

The homogeneous linear Fredholm integral equation of the second kind

\[ \xi(x,y)+(\nu+\alpha)\iint_D G(x,y;x_1,y_1) f'_u[x_1,y_1;u_\nu(x_1,y_1)]\xi(x_1,y_1)\,dx_1dy_1=0 \tag{14} \]

is equivalent to the Dirichlet problem for the equation

\[ E[\xi(x,y)]=(\nu+\alpha)f'_u[x,y;u_\nu(x,y)]\xi(x,y) \]

with zero data on the boundary \(\Gamma\) of the domain \(D\). Since \(f'_u(x,y,u)\ge 0\) everywhere in \(P\), the extremum principle holds for this problem. Therefore it cannot have solutions different from zero. Consequently, the equation (14) also cannot have nontrivial solutions, i.e. the number \(\nu+\alpha\) is not characteristic for the kernel

\[ G(x,y;x_1,y_1)f'_u[x_1,y_1;u_\nu(x_1,y_1)]. \]

Hence, in turn, there follows the existence of the resolvent \(R(x,y;x_1,y;\nu+\alpha)\), by means of which the nonlinear integral equation (12) is reduced to the equivalent integral equation

\[ \begin{aligned} \xi(x,y)&=-\alpha\iint_D f[x_1,y_1;u_\nu(x_1,y_1)] \{G(x,y;x_1,y_1)-\\ &\quad-(\nu+\alpha)\iint_D R(x,y;x',y';\nu+\alpha) G(x',y';x_1,y_1)\,dx'dy'\}\,dx_1dy_1\\ &\quad-\frac{1}{2}(\nu+\alpha)\iint_D \{f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\vartheta\xi(x_1,y_1)]\\ &\qquad\times\{G(x,y;x_1,y_1)-\\ &\quad-(\nu+\alpha)\iint_D R(x,y;x',y';\nu+\alpha) G(x',y';x_1,y_1)\,dx'dy'\}\,dx_1dy_1, \end{aligned} \]

or

\[ \begin{aligned} \xi(x,y)&=-\alpha\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha) f[x_1,y_1;u_\nu(x_1,y_1)]\,dx_1dy_1\\ &\quad-\frac{\nu+\alpha}{2}\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha) f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\vartheta\xi(x_1,y_1)]\\ &\qquad\times \xi^2(x_1,y_1)\,dx_1dy_1, \end{aligned} \tag{15} \]

where

\[ \Phi(x,y;x_1,y_1;\nu+\alpha)=G(x,y;x_1,y_1)- \]

\[ -(\nu+\alpha)\iint_D R(x,y;x',y';\nu+\alpha)G(x',y';x_1,y_1)\,dx_1dy_1 . \]

Since the function

\[ \Psi(x,y;\nu+\alpha)=\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha)\,dx_1dy_1 \]

is continuous for all \((x,y)\in \overline D\) and \(0\leq \nu+\alpha\leq 1\), there exists a constant \(B>0\) such that

\[ |\Psi(x,y;\nu+\alpha)|\leq B . \tag{16} \]

On the basis of (13) and (16) we shall have

\[ \left|\frac12(\nu+\alpha)\iint_D f''_{uu}\,[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \xi(x_1,y_1)]\xi^2(x_1,y_1)\times\right. \]

\[ \left.\times \Phi(x,y;x_1,y_1;\nu+\alpha)\,dx_1dy_1\right|\leq \]

\[ \leq \frac12(\nu+\alpha)\iint_D \left|f''_{uu}\,[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \xi(x_1,y_1)]\right| \cdot |\Phi(x,y;x_1,y_1;\nu+\alpha)|\times \]

\[ \times \xi^2(x_1,y_1)\,dx_1dy_1 \leq \frac12(\nu+\alpha)M_1B\left(\max_{\overline D}|\xi(x,y)|\right)^2\leq \]

\[ \leq M_1B\left(\max_{\overline D}|\xi(x,y)|\right)^2=B_1d^2, \tag{17} \]

where \(M_1=\max_{\overline P}|f''_{uu}(x,y,u)|\), and \(B_1=M_1B\).

Next, if \(\overline{\xi}(x,y)\) is an arbitrary function continuous in the closed domain \(\overline D\), for which the inequality

\[ \max_{\overline D}|\overline{\xi}(x,y)|\leq d \]

holds, then, by virtue of the assumptions made above concerning the derivative \(f''_{uu}(x,y,u)\), we obtain

\[ \left|\frac12(\nu+\alpha)\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha)\{f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\right. \]

\[ +\vartheta \xi(x_1,y_1)]\xi^2(x_1,y_1)-f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \overline{\xi}(x_1,y_1)]\overline{\xi}^{\,2}(x_1,y_1)\}\times \]

\[ \left.\times dx_1dy_1\right|\leq \iint_D |\Phi(x,y;x_1,y_1;\nu+\alpha)|\cdot \left|f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\right. \]

\[ +\vartheta \xi(x_1,y_1)]\xi^2(x_1,y_1) -f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \overline{\xi}(x_1,y_1)]\overline{\xi}^{\,2}(x_1,y_1)+ \]

\[ +f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \overline{\xi}(x_1,y_1)]\overline{\xi}^{\,2}(x_1,y_1)- \]

\[ -f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \overline{\xi}(x_1,y_1)]\overline{\xi}^{\,2}(x_1,y_1)\right|dx_1dy_1\leq \]

\[ \leq \iint_D |\Phi(x,y;x_1,y_1;\nu+\alpha)| \left\{\left|f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+\right.\right. \]

\[ +\vartheta \overline{\xi}(x_1,y_1)]\right|\cdot |\xi(x_1,y_1)+\overline{\xi}(x_1,y_1)|\cdot |\xi(x_1,y_1)-\overline{\xi}(x_1,y_1)|+ \]

\[ +\xi^2(x_1,y_1)\,L\vartheta\,|\zeta(x_1,y_1)-\overline{\zeta}(x_1,y_1)|\}\,dx_1dy_1 \le \]

\[ \le B\{2M_1d+L\vartheta d^2\}\max_{\overline D}|\zeta(x,y)-\overline{\zeta}(x,y)| = \]

\[ = B_2\bigl(\max_{\overline D}|\zeta(x,y)-\overline{\zeta}(x,y)|\bigr)\cdot d, \tag{18} \]

where \(B_2=(2M_1+\vartheta L)B\).

By virtue of (15) and (16), we have

\[ |\zeta(x,y)|\le \alpha BM+M_1B\bigl(\max_{\overline D}|\zeta(x,y)|\bigr)^2 = \]

\[ = \alpha BM+B_1\bigl(\max_{\overline D}|\zeta(x,y)|\bigr)^2 \]

for all \((x,y)\in\overline D\). Therefore

\[ \max_{\overline D}|\zeta(x,y)|\le \alpha BM+B_1\bigl(\max_{\overline D}|\zeta(x,y)|\bigr)^2, \]

or

\[ d\le \alpha BM+B_1d^2. \]

We shall be interested in such solutions \(\zeta(x,y)\) of the integral equation (15) for which \(\max_{\overline D}|\zeta(x,y)|\) does not exceed the smaller positive root

\[ \tau=\frac{1}{2B_1}\left(1-\sqrt{1-\alpha\cdot 4BB_1M}\right) \]

of the quadratic equation

\[ \tau=\alpha MB+B_1\tau^2. \]

The proof of the existence of \(\zeta(x,y)\) can be carried out by the method of successive approximations. Indeed,

\[ \zeta_1(x,y)=-\alpha\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha)\, f[x_1,y_1;u_\nu(x_1,y_1)]\,dx_1dy_1, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

\[ \zeta_k(x,y)=-\alpha\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha)\, f[x_1,y_1;u_\nu(x_1,y_1)]\,dx_1dy_1- \]

\[ -\frac{\nu+\alpha}{2}\iint_D \Phi(x,y;x_1,y_1;\nu+\alpha)\, f''_{uu}[x_1,y_1;u_\nu(x_1,y_1)+ \]

\[ +\vartheta\zeta_{k-1}(x_1,y_1)]\zeta_{k-1}^2(x_1,y_1)\,dx_1dy_1, \]

\[ \cdots\cdots\cdots\cdots\cdots\cdots\cdots\cdots \]

From inequalities (16) and (17) it follows that the estimates

\[ |\zeta_1(x,y)|\le \alpha BM<\tau, \]

\[ \cdots\cdots\cdots\cdots\cdots \]

\[ |\zeta_k(x,y)|\le \alpha BM+B_1\tau^2=\tau, \]

\[ \cdots\cdots\cdots\cdots\cdots \]

are valid; i.e., the functions \(\zeta_k(x,y)\), \(k=1,2,\ldots\), are uniformly bounded by the positive number \(\tau\).

It is easy to see that \(u_\nu(x,y)+\vartheta\zeta_k(x,y)\) belongs, for every \(k\), to the domain of definition of the function \(f(x,y,u)\). Further, by virtue of inequality (18), we have

\[ |\zeta_k(x,y)-\zeta_{k-1}(x,y)|\le \frac{1}{2}(\nu+\alpha)\iint_D |\Phi(x,y;x_1,y_1;\nu+\alpha)|\times \]

\[ \times \left| f''_{uu}\left[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \zeta_{k-1}(x_1,y_1)\right]\zeta_{k-1}^2(x_1,y_1) - \right. \]
\[ \left. - f''_{uu}\left[x_1,y_1;u_\nu(x_1,y_1)+\vartheta \zeta_{k-2}(x_1,y_1)\right]\zeta_{k-2}^2(x_1,y_1) \right|\,dx_1dy_1 \le \]
\[ \le B_2\tau\left(\max_{\overline D}|\zeta_{k-1}(x,y)-\zeta_{k-2}(x,y)|\right). \]

Thus,

\[ |\zeta_k(x,y)-\zeta_{k-1}(x,y)| \le B_2\tau\left(\max_{\overline D}|\zeta_{k-1}(x,y)-\zeta_{k-2}(x,y)|\right) \]

for any \((x,y)\in \overline D\) and \(k=1,2,\ldots\).

Hence we have

\[ \max_{\overline D}|\zeta_k(x,y)-\zeta_{k-1}(x,y)| \le (B_2\tau)^{k-1}\max_{\overline D}|\zeta_2(x_1,y_1)-\zeta_1(x_1,y_1)|. \]

Consequently, the series

\[ \zeta_1(x,y)+\sum_{k=1}^{\infty}[\zeta_{k+1}(x,y)-\zeta_k(x,y)] \tag{19} \]

converges for \(B_2\tau=q<1\). This condition will be satisfied if

\[ 0<\alpha<\frac{B_2-B_1}{MBB_2^2}=\alpha^*. \tag{20} \]

Note that the constants \(B, B_1, B_2\), and \(M\) do not depend on \(\nu,\alpha\). Subjecting \(\alpha\) to condition (20), the series (19) will converge absolutely and uniformly to some function \(\zeta(x,y)\), i.e., the sequence of functions \(\{\zeta_k(x,y)\}_{k=1,2,\ldots}\) converges absolutely and uniformly to the solution \(\zeta(x,y)\) of the integral equation (15), with \(|\zeta(x,y)|\le \tau\). Thus it has been proved that, under condition (20), the nonlinear integral equation (12) always has a solution.

Thus, assuming the existence of a solution of problem (8)—(2), we have shown the existence of a solution of problem (9)—(2) for \(\alpha\) satisfying condition (20).

Problem (3)—(2) for \(\lambda=0\), as was already said above, has a solution; i.e., we may proceed from the fact that problem (8)—(2) for \(\nu=0\) has a solution. Consequently, for some value \(\lambda=a_1<\alpha^*\) a solution of problem (9)—(2) exists.

Equating \(\nu\) to the number \(a_1\), we obtain that problem (3)—(2) has a solution for \(\lambda=a_1+a_2\), where \(a_2\) is some positive number smaller than \(\alpha^*\).

Repeating these arguments step by step, we arrive at \(\lambda=1\). Indeed, otherwise, beginning with some value

\[ \lambda=\sum a_j+\alpha=\nu+\alpha, \]

it would be impossible to repeat the arguments given above; i.e., \(\nu+\alpha\) would be a characteristic number for the kernel

\[ G(x,y;x_1,y_1) f'_u[x_1,y_1;u_\nu(x_1,y_1)], \]

and the linear homogeneous integral equation (14) would have a nontrivial solution. Consequently, the Dirichlet problem corresponding to equation (14) would also have a nonzero solution, which is impossible.

Theorem 2. There exists a unique solution \(u(x,y)\), twice continuously differentiable in the domain \(D\), of equation (1), continuous in the closed domain \(\overline D\) and satisfying the conditions

\[ u(x,y)\big|_{\sigma}=\varphi(s), \tag{21} \]

\[ \left. \frac{\partial u(x,y)}{\partial y}\right|_{AB}=\nu(x), \tag{22} \]

where \(\varphi(s)\) is a continuous function prescribed on \(\sigma\), and \(\nu(x)\) is also a prescribed continuous function on the interval \(-1<x<1\), which may tend to infinity of order less than \(\dfrac{2}{m+2}\) as \(x\to -1\) or \(x\to 1\).

The uniqueness of the solution of this problem is proved quite simply. Indeed, if we suppose that there exist two solutions \(u_1(x,y)\) and \(u_2(x,y)\), then their difference

\[ v(x,y)=u_1(x,y)-u_2(x,y) \]

will satisfy the equation

\[ E[v(x,y)] = f'_u[x,y;\,u_2+\vartheta(u_1-u_2)]\,v(x,y),\qquad 0<\vartheta<1 \tag{23} \]

with the boundary conditions

\[ \left. v(x,y)\right|_{\sigma}=0,\qquad \left. \frac{\partial v(x,y)}{\partial y}\right|_{AB}=0. \]

By virtue of the extremum principle \((f'_u(x,y;u)\geqslant 0)\) and the known Zaremba lemma (see, for example, [3]) we conclude that

\[ v(x,y)\equiv 0. \]

Hence, in turn, it follows that

\[ u_1(x,y)\equiv u_2(x,y). \]

We shall prove the existence of a solution of the boundary value problem (1)—(21)—(22) according to the scheme set forth above.

Denote by \(u_0(x,y)\) the solution of equation (4) satisfying the boundary conditions (21) and (22). Such a solution exists and is unique (see [4, 7]).

In the case of a normal domain the function \(u_0(x,y)\) is constructed explicitly in the form

\[ u_0(x,y)=\frac{1}{8\pi}\cos^2\beta\pi \left\{1-\frac{1}{\pi}\left[ \Gamma'(1)-\frac{\Gamma'(\beta)}{\Gamma(\beta)}\tg\beta\pi \right]\right\}^{-1}\times \]

\[ \times\left\{ -\int_{-1}^{1} N(x_1,0;\,x,y)\,\nu(x_1)\,dx_1 -\int_{-1}^{1}\left[ y_1^m\left( \frac{\partial N(x_1,y_1;\,x,y)}{\partial x_1}\frac{dy_1}{dx_1} -\frac{\partial N(x_1,y_1;\,x,y)}{\partial y_1} \right) \right]_{y_1=f(x_1)} \varphi(s)\,dx_1 \right\},\qquad (x,y)\in D, \]

where

\[ N(x,y;\,x_1,y_1)=\cos 2\beta\pi\, W(x,y;\,x_1,y_1)+\overline{W}(x,y;\,x_1,y_1)- \]

\[ -\left(x_1^2+\frac{4}{(m+2)^2}y^{m+2}\right)^{-\frac{m}{2m+2}} \left\{\cos 2\pi\beta\, W(x,y;\,\overline{x}_1,\overline{y}_1) +\overline{W}(x,y;\,\overline{x}_1,\overline{y}_1)\right\}, \]

\[ \beta=\frac{m}{2m+4}, \]

\[ \overline{x}_1=x_1\cdot\left(x_1^2+\frac{4}{(m+2)^2}\,y_1^{2+m}\right)^{-1}, \]

\[ \overline{y}_1=y_1^{\frac{m+2}{m}}\cdot\left(x_1^2+\frac{4}{(m+2)^2}\,y_1^{2+m}\right)^{-1}, \]

\[ W(x,y;x_1,y_1)= \left(\frac{4}{m+2}\right)^{m+2} r_1^{-\frac{m}{m+2}} \left\{ F\left(\beta,\beta,1;\frac{r^2}{r_1^2}\right)\log\frac{r^2}{r_1^2}+ \right. \]

\[ \left. +\left[ \left(\frac{\partial}{\partial a}+\frac{\partial}{\partial b}+2\frac{\partial}{\partial c}\right) F\left(a,b,c;\frac{r^2}{r_1^2}\right) \right]_{c=1,\ a=b=\beta} \right\}, \]

\[ \overline{W}(x,y;x_1,y_1)= \left(\frac{4}{m+2}\right)^{m+2} r^{-\frac{m}{m+2}} \left\{ F\left(\beta,\beta,1;\frac{r_1^2}{r^2}\right)\log\frac{r_1^2}{r^2}+ \right. \]

\[ \left. +\left[ \left(\frac{\partial}{\partial a}+\frac{\partial}{\partial b}+2\frac{\partial}{\partial c}\right) F\left(a,b,c;\frac{r_1^2}{r^2}\right) \right]_{c=1,\ a=b=\beta} \right\}, \]

\[ r^2=(x-x_1)^2+\frac{4}{(m+2)^2}\left[y^{\frac{m+2}{2}}-y_1^{\frac{m+2}{2}}\right], \]

\[ r_1^2=(x-x_1)^2+\frac{4}{(m+2)^2}\left[y^{\frac{m+2}{2}}+y_1^{\frac{m+2}{2}}\right], \]

\[ y=f(x)=\left[\left(\frac{m+2}{2}\right)^2(1-x^2)\right]^{\frac{1}{m+2}} \]
is the equation of the curve \(\sigma\).

The problem (3)—(21)—(22), with the aid of the Green’s function \(N(x,y;x_1,y_1)\), is reduced to the equivalent nonlinear integral equation

\[ u(x,y)=-\lambda\iint_D N(x,y;x_1,y_1)f[x_1,y_1;u(x_1,y_1)]\,dx_1dy_1+u_0(x,y), \tag{24} \]

the investigation of which is carried out analogously to equation (6).

References

1. Bitsadze A. V. Equations of mixed type. Publishing House of the Academy of Sciences of the USSR, Moscow, 1959.
2. Courant R. Partial differential equations. Mir Publishers, Moscow, 1964.
3. Miranda C. Partial differential equations of elliptic type. Foreign Literature Publishing House, Moscow, 1957.
4. Tricomi F. On linear partial differential equations of the second order of mixed type. OGIZ, Moscow—Leningrad, 1947.
5. Gellerstedt S. Sur un probleme aux limites pour une equation lineaire aux derivees partielles du second ordre de type mixte. These, Uppsala, 1935.
6. Gellerstedt S. Arkiv för Matematik, Astronomi och Fysik, 26A, No. 3, 1936.
7. Holmgren E. Arkiv för Matematik, Astronomi och Fysik, 19B, No. 14, 1926.
8. Lichtenstein L. Vorlesungen über einige Klassen nichtlinearer Integralgleichungen und Integrodifferentialgleichungen, Berlin, 1931.
9. Weinstein A. Transections of American Mathematical Society, t. 63, No. 2, 1948.

Received by the editors
September 30, 1965

Institute of Mathematics, Siberian Branch, Academy of Sciences of the USSR