UDC 517.946.9

THE DIRICHLET PROBLEM FOR DEGENERATING NONLINEAR ELLIPTIC EQUATIONS

M. I. Aliev

Consider the equation

\[ Gu \equiv y^m u_{yy}+u_{xx}-f(x,y,u,u_x,u_y)=0,\quad m>0, \tag{1} \]

in the domain \(D\), bounded by the segment \(AB\) of the \(x\)-axis and by a simple Jordan curve \(\sigma\), lying in the half-plane \(y>0\) and with endpoints at the points \(A\) and \(B\). It is elliptic inside the domain \(D\), while on the segment \(AB\) of its boundary it degenerates parabolically.

When \(f\) is a linear function with respect to \(u, u_x, u_y\), equation (1) was studied in detail by M. V. Keldysh [1], and it was established that, depending on the exponent \(m\) and the coefficient of \(u_y\), for equation (1) the Dirichlet problem, or the so-called problem \(E\), is solvable.

In the present work the Dirichlet problem for equation (1) is studied.

Find a solution of equation (1), twice continuously differentiable in the domain \(D\), continuous up to the boundary, and taking prescribed (not necessarily homogeneous, zero) boundary values.

The method used here is a certain modification of the Perron method [2], based on the construction of upper and lower functions. This method for nonlinear elliptic equations has also been used in works [3—6].

In what follows, the boundary of a domain \(\Delta\) will be denoted by \(\partial \Delta\).

We shall say that the function \(f\) satisfies condition \(A\), if:

1) \(f\) is a continuously differentiable function with respect to its arguments, and for \((x,y)\in D\) and for all \(u,u_x,u_y\) we have \(-f_u\leq -k<0\);

2) for \((x,y)\in D\) and \(|u|\leq N,\ -\infty<u_x,u_y<\infty\), the estimates \(|f_u|\leq M,\ |f_{u_x}|\leq M,\ |f_{u_y}|\leq M\) hold, where \(N\) and \(M(N)\) are some positive numbers.

The following obvious extremum principle holds.

The difference of two solutions \(u_1\) and \(u_2\) of equation (1) has neither a positive maximum nor a negative minimum inside the domain \(D\).

It follows from this that if a solution of the Dirichlet problem for equation (1) exists, then it is unique.

Let a continuous function \(\varphi(x,y)\) be given in the domain \(D\), and let some circle \(K\subset D\) be given. By \(\mathfrak{M}_K(\varphi)\) we denote the function which outside the circle \(K\) and on \(\partial K\) coincides with \(\varphi\), while inside it is the solution of equation (1) constructed from the boundary values \(\varphi\) on the circumference \(\partial K\).

The function \(\varphi\) will be called a supersolution (subsolution) if for every circle \(K\) lying in \(D\),

\[ \varphi\geq \mathfrak{M}_K(\varphi)\quad [\varphi\leq \mathfrak{M}_K(\varphi)]. \]

A supersolution (subsolution) \(\varphi\) will be called an upper (lower) func-

with respect to the function \(\gamma\), if it is continuous in \(\overline D=D+\partial D\) and \(\varphi \geqslant \gamma\) \((\varphi \leqslant \gamma)\) on \(\partial D\).

We shall prove several auxiliary lemmas.

Lemma 1. If in the domain \(D\), for some function \(\varphi(x,y)\), one has \(G\varphi \leqslant 0\) \((G\varphi \geqslant 0)\), then for any circle \(K\subset D\) one has
\[ \mathfrak M_K(\varphi)\leqslant \varphi \quad [\mathfrak M_K(\varphi)\geqslant \varphi]. \]

We prove by contradiction. Suppose the lemma is false. Then there exists a circle \(K\) such that the solution \(v\), constructed in this circle from the boundary values \(\varphi\), is greater (less) than \(\varphi\). The difference \(w=v-\varphi\) is equal to zero on \(\partial K\). By the continuity of \(w\), at some interior point \(P\) it assumes a positive maximum (negative minimum).

Applying the mean-value theorems, we obtain
\[ Gv-G\varphi = y^m w_{yy}+w_{xx}-\overline f_w w-\overline f_{w_x}w_x-\overline f_{w_y}w_y, \]
where
\[ \overline g = g[x,y,\varphi+\theta(v-\varphi),\varphi_x+\theta_1(v_x-\varphi_x), \varphi_y+\theta_2(v_y-\varphi_y)]. \]

Hence, taking into account that at the point \(P\) \(w\) attains a positive maximum, we obtain that \(Gv-G\varphi<0\) \((Gv-G\varphi>0)\). This contradicts the fact that \(Gv=0\) and \(G\varphi\leqslant0\) \((G\varphi\geqslant0)\).

The lemma is proved.

It follows from this lemma that if a twice continuously differentiable function \(\varphi\) is a supersolution, then \(G\varphi\leqslant0\), and if it is a subsolution, then \(G\varphi\geqslant0\). The solution \(u\), however, may be regarded both as a supersolution and as a subsolution.

Lemma 2. Every upper function \(\varphi\) is nowhere smaller than every lower function \(\psi\).

Suppose that at some interior point \(P_0\) the difference \(\varphi-\psi\) attains a negative minimum. Take some circle \(K\) with center at the point \(P_0\). Taking into account
\[ \varphi\geqslant \mathfrak M_K(\varphi),\qquad \psi\leqslant \mathfrak M_K(\psi), \]
we have
\[ (\varphi-\psi)_{P_0}\geqslant [\mathfrak M_K(\varphi)-\mathfrak M_K(\psi)]_{P_0}. \]

By assumption,
\[ (\varphi-\psi)_{P_0}\leqslant \min_{P\in\partial K}(\varphi-\psi) = \min_{P\in\partial K}[\mathfrak M_K(\varphi)-\mathfrak M_K(\psi)]. \]

Consequently, we obtain that
\[ [\mathfrak M_K(\varphi)-\mathfrak M_K(\psi)]_{P_0} \leqslant \min_{P\in\partial K}[\mathfrak M_K(\varphi)-\mathfrak M_K(\psi)]. \]

But the difference of two solutions cannot attain a negative minimum in the interior. The contradiction proves the lemma.

Lemma 3. If \(\varphi_1,\varphi_2,\ldots,\varphi_n\) are upper (lower) functions, then
\[ \varphi=\min(\varphi_1,\varphi_2,\ldots,\varphi_n) \quad [\varphi=\max(\varphi_1,\varphi_2,\ldots,\varphi_n)] \]
is also an upper (lower) function.

For the proof it is enough to show that, for any circle \(K\subset D\),
\[ \varphi\geqslant \mathfrak M_K(\varphi) \quad [\varphi\leqslant \mathfrak M_K(\varphi)]. \]
At any point \(P\in D\), the value \(\varphi(P)\) is equal to the value at this point of one of the functions \(\varphi_1,\varphi_2,\ldots,\varphi_n\), say \(\varphi_1\). Therefore at the point \(P\)
\[ \varphi=\varphi_1\geqslant \mathfrak M_K(\varphi_1)\geqslant \mathfrak M_K(\varphi) \quad [\varphi=\varphi_1\leqslant \mathfrak M_K(\varphi_1)\leqslant \mathfrak M_K(\varphi)], \]
since if \(\varphi\leqslant\psi\), then always \(\mathfrak M_K(\varphi)\leqslant \mathfrak M_K(\psi)\).

Lemma 4. If \(\varphi\) is an upper (lower) function, then \(v=\mathfrak M_K(\varphi)\) is also an upper (lower) function.

It is enough to show that for any disk \(K_1\subset D\) one has \(v\geq \mathfrak M_{K_1}(v)\) \([v\leq \mathfrak M_{K_1}(v)]\). This is obvious if \(K_1\) lies entirely outside or entirely inside the disk \(K\). It is also obvious when \(K\) lies inside \(K_1\).

Now suppose the disks intersect. In this case, if the point \(P\) lies outside \(K_1\), then \(v\) and \(\mathfrak M_K(v)\) coincide, while for points \(P\in K_1-K\) we have

\[ v=\mathfrak M_K(\varphi)=\varphi\geq \mathfrak M_{K_1}(\varphi)\geq \mathfrak M_{K_1}(v) \quad [v=\mathfrak M_K(\varphi)=\varphi\leq \mathfrak M_{K_1}(\varphi)\leq \mathfrak M_{K_1}(v)] . \]

Finally, let \(P\in K\cap K_1\). On the part of the boundary \(\partial(K\cap K_1)\) lying inside \(K\), we have \(v-\mathfrak M_K(v)=0\), and on the part of the boundary lying inside \(K_1\), we shall have

\[ v-\mathfrak M_{K_1}(v)\geq \varphi-\mathfrak M_{K_1}(\varphi)\geq 0 \quad [v-\mathfrak M_{K_1}(v)\leq \varphi-\mathfrak M_{K_1}(\varphi)\leq 0]. \]

Consequently, on the boundary \(\partial(K\cap K_1)\) the difference \(v-\mathfrak M_{K_1}(v)\) is nonnegative (nonpositive). But in the domain \(K\cap K_1\), \(v\) and \(\mathfrak M_{K_1}(v)\) are solutions of equation (1). By the extremum principle we conclude that also inside \(K\cap K_1\) one has \(v-\mathfrak M_{K_1}(v)\geq 0\) \([v-\mathfrak M_{K_1}(v)\leq 0]\). This completes the proof of the lemma.

Harnack’s theorem holds. If, under condition \(A\), a sequence of solutions of equation (1) converges uniformly on the boundary \(\partial\Omega\) of a domain \(\Omega\) lying in \(D\), then it converges uniformly inside \(\Omega\) as well, to a solution of equation (1).

Indeed, if \(\{u_n\}\) is a sequence of solutions uniformly converging on the boundary \(\partial\Omega\), then the difference \(u_m-u_n\) attains its positive maximum and negative minimum on \(\partial\Omega\). But this difference on \(\partial\Omega\) is arbitrarily small. Hence the uniform convergence of \(\{u_n\}\) in \(\Omega\) and the continuity of the limiting function in \(\Omega+\partial\Omega\) follow.

Let \(\{u_n\}\) tend on the boundary \(\partial\Omega\) to the function \(\varphi\). Since under condition \(A\) equation (1), considered in the domain \(\Omega\), belongs to Bernstein’s class \(L\) (see [7]), there exists a solution \(u\) of equation (1) in \(\Omega\), taking the values \(\varphi\) on \(\partial\Omega\) (see also [8], pp. 115–128). But

\[ \max_{\overline{\Omega}} |u-u_n|\leq \max_{\partial\Omega}|\varphi-\varphi_n|<\varepsilon . \]

Consequently, the sequence \(\{u_n\}\) tends to the solution \(u\) in \(\Omega\).

A twice continuously differentiable (with respect to the variables \((x,y)\)) function \(w(x,y;x_0,y_0)\), having the properties: 1) \(w(x,y;x_0,y_0)>0\) everywhere in \(D\); 2) \(w(x_0,y_0;x_0,y_0)=0\) at the point \(P_0(x_0,y_0)\in\partial D\); 3) \(Gw\leq 0\) in \(D\), will be called a barrier at the point \(P_0\).

Denote by \(D_h\) the domain consisting of points of the domain \(D\) satisfying the condition \(y>h\), where \(h\) is a sufficiently small positive number.

One of the following conditions will be called condition \(B\): a) \(m<1\); b) when \(m=1\),

\[ -\frac{\partial f}{\partial u_y}<1 \]

in the domain \(\mathcal R_h=\{(x,y)\in D-D_h,\ |u|\leq N,\ -\infty<u_x,u_y<\infty\}\); c) when \(1<m<2\),

\[ -\frac{\partial f}{\partial u_y}<y^{m-1} \]

in the domain \(\mathcal R_h\); d) when \(m\geq 2\),

\[ -\frac{\partial f}{\partial u_y}<0 \]

in the domain \(\mathcal R_h\).

The following holds.

Lemma 5. Suppose that conditions \(A\) and \(B\) are fulfilled for equation (1). Let the curve \(\sigma\) be such that for each of its points \(P_0\) there exists a disk \(S_{P_0}\) whose intersection with \(D+\partial D\) consists of the single point \(P_0\). Then at every point \(P_0\) of the boundary \(\partial D\) there exists a barrier.

Suppose that the point \(P_0\) lies on the line of degeneration. Denote by \(\mu\) the number

\[ \mu=\frac{\max_{\overline D}|f(x,y,0,0,0)|}{k}, \tag{2} \]

where \(k\) is the number from condition \(A\), and consider the function

\[ w_0(x,y;\xi,0)=\mu-\frac{1}{\left[(x-\xi)^2+\mu^{-1/p}(y^\beta+1)\right]^p}, \]

where \(0<\beta<1\), \(p>0\) is a numerical parameter, and \(\xi\) is the abscissa of the point \(P_0\).

We shall show that for large values of the parameter \(p\) the function \(w_0\) will be a barrier at the point \(P_0\).

Indeed, the fulfillment of conditions 1) and 2) is obvious. Let us verify 3). First note that \(0<w_0<\mu\) for any value of the parameter \(p\). Further, we have

\[ Gw_0=-\frac{\beta\mu^{-1/p}py^{\beta-1}}{D^{p+1}} \left[-(\beta-1)y^{m-1}+\overline f_{w_0y}\right]-- \]

\[ -\frac{p}{D^{p+1}} \left\{ \frac{p+1}{D}\left[\beta^2\mu^{-2/p}y^{m-2+2\beta}+4(x-\xi)^2\right] -2+2\overline f_{w_0x}(x-\xi) \right\} -\overline f_{w_0}w_0-f_0, \]

where

\[ D=(x-\xi)^2+\mu^{-1/p}(y^\beta+1), \]

\[ \overline g=g(x,y,\theta u,\theta_1u_x,\theta_2u_y),\qquad f_0=f(x,y,0,0,0). \]

In the domain \(D-D_h\), for sufficiently small \(h>0\), the inequality \(Gw_0\le 0\) is ensured by conditions \(A\) and \(B\). In the domain \(D_h\) this is achieved by the choice of the parameter \(p\), since for large \(p\) we have

\[ Gw_0< \frac{p}{D^{p+1}} \left\{ -\frac{p+1}{D}\left[\beta^2\mu^{-2/p}y^{m-2+2\beta}+4(x-\xi)^2\right]+2-2\overline f_{w_0x}(x-\xi)+ \right. \]

\[ \left. +\beta(\beta-1)\mu^{-1/p}y^{m-2+\beta} -\beta\mu^{-1/p}y^{\beta-1}\overline f_{w_0y} \right\}<0. \]

Now suppose that the point \(P_0(x_0,y_0)\) lies on \(\sigma\) (we regard the curve \(\sigma\) as open). Denote the center of the circle \(S_{P_0}\) by \((\overline x,\overline y)\) and consider the function

\[ w_1(x,y;x_0,y_0)=\mu(1+R^2)^{p_1}\Omega(x,y;x_0,y_0)(w_0(x,y;\xi,0)+1), \]

where \(\mu\) is the number defined by (2), \(p_1>0\) is a numerical parameter, and

\[ \Omega(x,y;x_0,y_0)=\frac{1}{(1+R^2)^{p_1}}-\frac{1}{(1+r^2)^{p_1}}, \]

\[ R^2=(x_0-\overline x)^2+(y_0-\overline y)^2,\qquad r^2=(x-\overline x)^2+(y-\overline y)^2. \]

We shall prove that for sufficiently large values of the parameter \(p_1\) the function \(w_1\) will be a barrier at the point \(P_0\). It is obvious that for all values of \(p_1\) we have
\(0<w_1<\mu(\mu+1)\).

Properties 1) and 2) are obvious. Further,

\[ Gw_1=\mu(1+R^2)^{p_1} \left\{(y^m\Omega_{yy}+\Omega_{xx})(w_0+1)+[-\overline f_{w_1x}(w_0+1)+\right. \]

\[ \left. +2w_{0x}]\Omega_x+[-\overline f_{w_1y}(w_0+1)+2y^mw_{0y}]\Omega_y +(G_1w_0-\overline f_{w_1})\Omega \right\}-f_0, \]

where

\[ G_1 w_0 = y^m w_{0yy} + w_{0xx} - \bar f_{w_{1x}} w_{0x} - \bar f_{w_{1y}} w_{0y} + \bar f_{w_1} w_0, \]

or

\[ G w_1 \leq \mu (1+R^2)^{p_1} \left\{ (y^m \Omega_{yy}+\Omega_{xx})(w_0+1)+E\Omega_x+F\Omega_y+ \right. \]

\[ \left. +\,G_1 w_0\,\Omega+\frac{\bar f_{w_1}}{(1+r^2)^{p_1}} \right\}, \]

where

\[ E=-\bar f_{w_{1x}}(w_0+1)+2w_{0x},\qquad F=-\bar f_{w_{1y}}(w_0+1)+2y^m w_{0y}. \]

When \(y \to 0\), the coefficient of \(\Omega\) tends to \(-\infty\) at a higher order than the other terms. Therefore in \(D-D_h\) we shall have \(G w_1 \leq 0\), while in \(D_h\) this is achieved by choosing \(p_1\) large, since

\[ G w_1 \leq \mu (1+R^2)^{p_1} \left\{ -\frac{4(p_1+1)}{1+r^2} \bigl[y^m(y-\bar y)^2+(x-\bar x)^2\bigr](w_0+1) \right. \]

\[ \left. +\,2(w_0+1)(y^m+1)+2E(x-\bar x)+2F(y-\bar y) +\frac{\bar f_{w_1}}{p_1}(1+r^2) \right\}\leq 0 \]

for large \(p_1\).

Lemma 5 is proved.

Lemma 6. If at each point \(P_0\) of the boundary \(\partial D\) there exists a barrier \(w(x,y;x_0,y_0)\), then one can construct some upper function \(\varphi_{P_0}\) and lower function \(\psi_{P_0}\) relative to the boundary function \(\gamma=0\), taking at the point \(P_0\) the values \(\varepsilon/2\) and \(-\varepsilon/2\), respectively, where \(\varepsilon>0\) is any prescribed number.

Indeed, take the function

\[ \varphi_{P_0}=\frac{\varepsilon}{2}+w(x,y;x_0,y_0), \]

where, for points on the line of degeneracy, \(w\) is understood as \(w_0(x,y;\xi,0)\), and for points of \(\sigma\) we understand \(w_1(x,y,x_0,y_0)\). The function \(\varphi_{P_0}\) will be the upper function we need.

Indeed, \(\varphi_{P_0}>0\), \(\partial D\), and \(\varphi_{P_0}(P_0)=\dfrac{\varepsilon}{2}\). It remains to show that \(\varphi\) is a supersolution; and by Lemma 1 it is enough for this to show that \(G\varphi_{P_0}\leq 0\). But this is proved as above.

Similarly one can show that the function

\[ \psi_{P_0}=-\frac{\varepsilon}{2}-w(x,y;x_0,y_0) \]

is a lower function and \(\psi_{P_0}(P_0)=-\dfrac{\varepsilon}{2}\).

From the continuity of the upper and lower functions it follows that on the contour \(\partial D\) there exists some neighborhood of the point \(P_0\) on which \(\varphi_{P_0}\) and \(\psi_{P_0}\) differ from each other by \(2\varepsilon\). By the Heine–Borel lemma we choose a finite number of neighborhoods covering the whole contour \(\partial D\). To these neighborhoods there corresponds a finite number of upper functions \(\varphi_1,\ldots,\varphi_n\) and lower functions \(\psi_1,\ldots,\psi_n\). By Lemma 3 the functions \(\varphi=\min(\varphi_1,\ldots,\varphi_n)\) and \(\psi=\max(\psi_1,\ldots,\psi_n)\) will also be upper and lower functions, respectively.

Since \(\varphi\) is an upper function, at any point \(P \in \partial D\) we have \(\varphi(P) \geq 0\). On the other hand, at the point \(P\) the value of \(\varphi\) is equal to one of the values \(\varphi_i\): \(\varphi(P)=\varphi_i(P)\leq 2\varepsilon\). Therefore, \(0\leq \varphi(P)\leq 2\varepsilon\).

Similarly, we obtain that \(-2\varepsilon \leq \psi(P)\leq 0\). This proves the following

Lemma 7. There exist an upper function \(\varphi\) and a lower function \(\psi\), which on the entire contour differ from the values of the boundary function \(\gamma(P)=0\) by less than any \(\varepsilon>0\).

It follows from this that the upper envelope of the lower functions and the lower envelope of the upper functions everywhere on the contour \(\partial D\) coincide with the prescribed boundary function \(\gamma(P)=0\).

Lemma 8. The lower envelope of the upper functions coincides with the upper envelope of the lower functions everywhere in \(D\) and is a solution of equation (1).

If we construct an upper function \(\tilde{\varphi}\) and a lower function \(\tilde{\psi}\), whose difference \(\tilde{\varphi}-\tilde{\psi}\) at every point \(P \in D\) is less than a prescribed \(\varepsilon>0\), then from this the proof of the first part of the lemma will follow.

Suppose that the difference of the functions \(\varphi\) and \(\psi\), constructed according to Lemma 7, is not small inside the domain \(D\). Then the difference \(\varphi-\psi\) attains a positive maximum at some interior point \(Q\). Denote this maximum by \(A\).

Let \(D_1\) be an arbitrary domain lying in \(D\). We may assume that \(Q\) belongs to \(D_1\) and, consequently, to one of the disks \(K_i\) covering the domain \(D_1\) and lying in \(D\).

Construct an upper function \(\varphi^{(1)}\) and a lower function \(\psi^{(1)}\) such that

\[ \max_{\overline{D}_1}\bigl(\varphi^{(1)}-\psi^{(1)}\bigr)<qA, \tag{3} \]

where \(0<q<1\) is some number independent of \(A\).

Using \(\varphi\) and \(\psi\), form the new functions

\[ \varphi_i=\mathfrak{M}_{K_i}(\varphi), \qquad \psi_i=\mathfrak{M}_{K_i}(\psi). \]

By Lemma 4, \(\varphi_i\) and \(\psi_i\) are upper and lower functions.

Take a disk \(L_i\), smaller than and concentric with the disk \(K_i\), and containing the point \(Q\). Clearly,

\[ \max_{L_i}(\varphi_i-\psi_i)\leq \max_{K_i}(\varphi_i-\psi_i). \]

But we shall show that

\[ \max_{L_i}(\varphi_i-\psi_i)\leq q_i A, \tag{4} \]

where \(0<q_i<1\) is some number depending only on \(L_i\).

In the disk \(K_i\), the difference \(\chi=\varphi_i-\psi_i\) satisfies the equation

\[ F\chi \equiv y^m \frac{\partial^2 \chi}{\partial y^2} + \frac{\partial^2 \chi}{\partial x^2} - \frac{\partial \tilde f}{\partial \chi_x}\frac{\partial \chi}{\partial x} - \frac{\partial \tilde f}{\partial \chi_y}\frac{\partial \chi}{\partial y} - \frac{\partial \tilde f}{\partial \chi}\chi = 0 . \]

The function

\[ \tilde{\chi}= \left[ \exp \alpha(r^2-R^2)+\lambda-\frac{\lambda}{(y+1)^\nu} \right]A \]

with a suitable choice of the parameters \(\alpha>0\) and \(\nu>0\) will be an upper function in the disk \(K_i\) relative to the boundary values \(\varphi_i-\psi_i\) for the equation \(F\chi=0\), where
\(r^2=(x-x_1)^2+(y-y_1)^2\), \((x_1,y_1)\) is the center of the disk \(K_i\) of radius \(R\), and \(0<\lambda<1\).

Indeed, on \(\partial K_i\) we have \(\tilde{\chi}\geq \varphi_i-\psi_i\). It remains to show that \(\tilde{\chi}\) is a supersolution for the equation \(F\chi=0\).

We have

\[ F\widetilde X=2\alpha\exp\alpha(r^2-R^2)A\,[y^m+2\alpha(y-y_1)^2y^m+1+2\alpha(x-x_1)^2- \]

\[ -\widetilde f_{\widetilde X_y}(y-y_1)-\widetilde f_{\widetilde X_x}(x-x_1) +\frac{\lambda\nu A}{(y+1)^{\nu+1}}\left[-\frac{y^m}{1+y}(\nu+1)-\widetilde f_{\widetilde X_y}\right]-\widetilde f_X\widetilde X . \]

From this it is easy to see that for sufficiently small \(\alpha\) and large \(\nu\), \(F\widetilde X\leqslant 0\). This proves that \(\widetilde X\) is a supersolution for the equation \(FX=0\).

After choosing \(\alpha\), we choose \(\lambda\) from the inequality

\[ \exp\alpha(\rho^2-R^2)+\lambda<1, \tag{5} \]

where \(\rho\) is the radius of the circle \(L_i\). Since \(\rho<R\), such a \(\lambda\) always exists.

Since \(\widetilde X\) is an upper function in the circle \(K_i\), inside it
\(\varphi_i-\psi_i<\widetilde X\). Further,

\[ \max_{L_i}(\varphi_i-\psi_i)<\max_{L_i}\widetilde X<\max_{\partial L_i}\widetilde X= \]

\[ =\left[\exp\alpha(\rho^2-R^2)+\lambda-\frac{\lambda}{(Y+1)^\nu}\right]A=q_iA, \]

where by \(Y\) we have denoted the maximum of the ordinates of the points of the circumference \(\partial L_i\). Thus, taking (5) into account, estimate (4) is proved.

The domain \(\overline D_1\) can be covered by a finite number \(n\) of circles \(L_i\), internal with respect to the domain \(D\). Carrying out this operation for each circle \(L_i\), we obtain \(\varphi_1,\ldots,\varphi_n\) upper functions and \(\psi_1,\ldots,\psi_n\) lower functions. Let

\[ \varphi^{(1)}=\min(\varphi_1,\ldots,\varphi_n),\qquad \psi^{(1)}=\max(\psi_1,\ldots,\psi_n). \]

By Lemma 3, \(\varphi^{(1)}\) and \(\psi^{(1)}\) will be upper and lower functions.

Since each point of the domain \(\overline D_1\) belongs to some circle \(L_j\), we have

\[ \max_{\overline D_1}(\varphi^{(1)}-\psi^{(1)}) \leqslant \max_{L_j}(\varphi_j-\psi_j)<q_jA\leqslant qA, \]

where \(q=\max(q_1,\ldots,q_n)\). We obtain estimate (3). Continuing this process, we construct sequences of upper functions \(\varphi^{(1)},\varphi^{(2)},\ldots\) and lower functions \(\psi^{(1)},\psi^{(2)},\ldots\), such that

\[ \varphi^{(l)}-\psi^{(l)}<q^lA. \]

Since \(q<1\), and the sequence of domains \(D_1\) can contain any point of the domain \(D\), from this we obtain the proof of the first part of the lemma, i.e., the upper envelope of the lower functions everywhere in \(D\) coincides with the lower envelope of the upper functions. Denote it by \(v(x,y)\).

It remains to show that \(v(x,y)\) is a solution of equation (1). For this it is enough to prove that for any circle \(K\subset D\), \(v=\mathfrak M_K(v)\).

From the constructed upper functions \(\varphi^{(1)},\varphi^{(2)},\ldots\) form the functions \(v_n=\mathfrak M_K(\varphi^{(n)})\). The sequence \(\{v_n\}\) is a sequence of upper functions (by Lemma 4). \(\{\varphi^{(n)}\}\) at every point of the domain \(D\) converges uniformly to its lower envelope. Outside the circle \(K\) and on \(\partial K\), \(v_n\) coincides with \(\varphi^{(n)}\) and, consequently, converges uniformly to the function \(v\). Inside \(K\), \(\{v_n\}\) is a sequence of solutions uniformly convergent on the boundary \(\partial K\). By Harnack’s theorem, the sequence \(\{v_n\}\) inside \(K\) converges uniformly to a solution of equation (1). From the continuity of this limiting function in \(K+\partial K\) it follows that \(v=\mathfrak M_K(v)\).

Lemma 8 is completely proved.

Thus, we have proved the following theorem.

Theorem. Suppose that for equation (1) conditions \(A\) and \(B\) are satisfied. Let the curve \(\sigma\) be such that it satisfies the condition of Lemma 5. Then in the domain \(D\) the Dirichlet problem for equation (1) always has, moreover, a unique solution.

References

1. Keldysh M. V. Dokl. Akad. Nauk SSSR, 77, No. 2, 181—183, 1951.
2. Perron O. Math. Z., 18, 42—54, 1923.
3. Simonov N. I. Bulletin of Moscow State University, 2, issue 1, 1939.
4. Akô K. J. Fac. Sci. Univ. Tokyo, sec. 1, 9, part. 3, 165—202, 1962.
5. Akô K. J. Fac. Sci. Univ. Tokyo, sec. 1, 9, part. 5, 383—395, 1963.
6. Akô K. J. Fac. Sci. Univ. Tokyo, sec. 1, 10, part. 2, 81—87, 1964.
7. Bernstein S. N. Collected Works, vol. III. Moscow, Publishing House of the Academy of Sciences of the USSR, 1960.
8. Bakelman N. Ya. Geometric Methods for Solving Elliptic Equations. Moscow, “Nauka” Publishing House, 1965.

Received by the editors
May 6, 1966

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