A. A. VASHARIN and P. I. LIZORKIN

CERTAIN BOUNDARY-VALUE PROBLEMS FOR ELLIPTIC EQUATIONS WITH STRONG DEGENERATION AT THE BOUNDARY

(Presented by Academician S. L. Sobolev, 25 XI 1960)

For the equation

\[ \frac{\partial}{\partial x}\left[y^k\frac{\partial u}{\partial x}\right] + \frac{\partial}{\partial y}\left[y^k\frac{\partial u}{\partial y}\right] =0, \tag{1} \]

considered in a bounded simply connected domain \(G\) with piecewise smooth boundary \(\Gamma\), containing a segment \(\Gamma_0\) of the axis \(Ox\), the formulation of boundary-value problems depends essentially on the value of \(k\). From the results of M. V. Keldysh \(\left({}^{1}\right)\) it follows that, in the class of continuous boundary data and solutions bounded in \(\overline{G}\), for equation (1) either the Dirichlet problem is uniquely solvable when \(k<1\), or problem E is uniquely solvable when \(k>1\) (in which the segment \(\Gamma_0\) is freed from boundary conditions). Analogous results were obtained by other authors (see the monograph \(\left({}^{2}\right)\) and the bibliography therein), proceeding from a variational treatment of the corresponding problems. In the present paper we investigate other formulations of boundary-value problems for equations of type (1), originating with A. V. Bitsadze \(\left({}^{3}\right)\).

In a domain \(G\) of the indicated kind, consider the equation

\[ L(u)= \frac{\partial}{\partial x}\left[\sigma^k(x,y)\frac{\partial u}{\partial x}\right] + \frac{\partial}{\partial y}\left[\sigma^k(x,y)\frac{\partial u}{\partial y}\right] =0, \tag{2} \]

where \(\sigma(x,y)\) is a sufficiently smooth positive function satisfying the estimate

\[ c_1y<\sigma(x,y)<c_2y,\qquad c_1,c_2>0. \]

Thus equation (2) degenerates on the segment \(\Gamma_0\). We shall call the degeneration strong (weak) for \(k\geq 1\) \((k<1)\); the case \(k=1\) will also be called critical. Let first \(k>1\).

Problem A. Find a solution \(u(x,y)\), twice continuously differentiable in the domain \(G\), of equation (2), which assumes on the boundary \(\Gamma\), in the mean, the values

\[ \lim_{(x,y)\to M\in\Gamma}\left[\sigma^{k-1}(x,y)u(x,y)\right]=\varphi(M). \tag{3} \]

Theorem 1. Suppose that \(\Gamma\) does not touch the axis \(Ox\) and contains no reentrant corners. If \(1<k<2\), the function \(\sigma(x,y)\) is four times boundedly differentiable and \(\Delta\sigma\geq 0\), then problem A has a unique solution provided the requirements

\[ \text{a) }\ \varphi(M)\in L_2(\Gamma);\qquad \text{b) }\ \int_{\Gamma} ds_M \int_{\Gamma} \frac{|\varphi(M)-\varphi(Q)|^2}{|MQ|^2} \,\omega^{\,2-k}(M,Q)\,ds_Q<\infty, \tag{4} \]

are fulfilled, where \(\omega(M,Q)\) is equal to the distance \(|MQ|\) between the points \(M\) and \(Q\), if at least one of these points lies on \(\Gamma_0\), and is equal to the lesser of the distances from these points to the axis \(Ox\) otherwise.

For the proof, consider the auxiliary functional

\[ I(v)=\iint\limits_G\left\{\sigma^{2-k}\left[\left(\frac{\partial v}{\partial x}\right)^2+\left(\frac{\partial v}{\partial y}\right)^2\right]+(k-1)\Delta\sigma\cdot\sigma^{1-k}v^2\right\}\,dx\,dy \tag{5} \]

in the class \(W(\varphi)\) of functions from \(W^{(1)}_{2,\,2-k}\) (see \((^2)\)), taking on the boundary the prescribed values \(\varphi(M)\). Conditions (4) constitute a characterization (a necessary and sufficient criterion) of the boundary values of functions from \(W^{(1)}_{2,\,2-k}\), and therefore the class \(W(\varphi)\) is nonempty. On the basis of arguments customary for the variational method \((^4)\), we are convinced of the existence of a unique element \(v_0\in W(\varphi)\) minimizing the functional \(I(\varphi)\). The function \(v_0\), by virtue of the conditions of the theorem, turns out to be twice continuously differentiable \((^2)\), satisfies the Euler–Lagrange equation of the functional (5), and, in the mean quadratic sense, the boundary condition

\[ \lim_{(x,y)\to M\in\Gamma} v_0(x,y)=\varphi(M). \]

From the uniqueness theorem of paper \((^5)\) (p. 263) it follows that there can be no other solution with the indicated properties. By a direct verification one may now ascertain that the function \(u_0=v_0/\sigma^{k-1}\) satisfies equation (2) and is a solution of Problem A. It may also be asserted that for the function \(u_0\) the integral \(I(\sigma^{k-1}u_0)\) is finite.* The theorem is proved.

Let us note that the restrictions \(k<2\) and those on \(\sigma\) have been imposed here by the method of proof. Problem A has meaning also under broader assumptions, in particular in the case of strong degeneration along the entire boundary. As a useful example, consider Problem A for equation (1) in the half-plane \(y>0\). After the substitution \(y^{k-1}u=v\), equation (1) takes the form

\[ \frac{\partial}{\partial x}\left(y^{2-k}\frac{\partial v}{\partial x}\right)+ \frac{\partial}{\partial y}\left(y^{2-k}\frac{\partial v}{\partial y}\right)=0, \]

and the solution of the posed problem for all \(k>1\) is given by the formula

\[ u_0(x,y)=\frac{1}{\sqrt{\pi}}\, \frac{\Gamma(k/2)}{\Gamma((k-1)/2)} \int_{-\infty}^{\infty} \frac{\varphi(\xi)\,d\xi}{[(x-\xi)^2+y^2]^{k/2}} . \tag{6} \]

The uniqueness of this solution for \(1<k<3\) can be obtained from the requirement that \(D_{2-k}(y^{k-1}u_0)\) be finite; here the boundary function \(\varphi(x)\) must be taken from \(W^{(k-1)/2}_2(-\infty,\infty)\). These assertions follow immediately from the results of paper \((^6)\).

The case of critical degeneration \(k=1\) is of interest. For simplicity we shall restrict ourselves to the case \(\sigma=y\) and pose the problem as follows:

Problem B. Find in the domain \(G\) a twice continuously differentiable solution of the equation

\[ L_1(u)=y\left(\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}\right)+\frac{\partial u}{\partial y}=0, \]

which takes on the boundary of the domain, in the mean sense, the values

\[ \left[ \frac{1}{\ln \dfrac{M}{y}}\,u(x,y) \right]_{(x,y)\to P\in\Gamma} =\varphi(P) \]

and has finite integral

\[ \iint\limits_G y\ln^2\frac{M}{y} \left\{ \left[ \frac{\partial}{\partial x}\left(\frac{1}{\ln \dfrac{M}{y}}\,u\right) \right]^2 + \left[ \frac{\partial}{\partial y}\left(\frac{1}{\ln \dfrac{M}{y}}\,u\right) \right]^2 \right\}\,dx\,dy, \]

where \(M\) is a constant exceeding the diameter of the domain.

\[ \text{* If one requires in advance of the solution the finiteness of the integral } I(\sigma^{k-1}u), \text{ then conditions (4) for the unique solvability of Problem A are not only sufficient but also necessary.} \]

Let the domain \(G\) satisfy the same requirements as in Theorem 1, and let \(\omega(P,Q)\) have the same meaning.

Theorem 2. For Problem B to have a solution, it is necessary and sufficient that the boundary function \(\varphi(P)\) satisfy the conditions

\[ \text{a) }\ \varphi(P)\in L_2(\Gamma);\qquad \text{b) }\ \int_\Gamma d s_P \int_\Gamma \frac{|\varphi(P)-\varphi(Q)|^2}{|PQ|^2}\,\omega(P,Q)\,d s_Q<\infty . \tag{7} \]

Proof. Consider the functional

\[ D(v,G)=\iint_G y\ln^2\frac{M}{y} \left[\left(\frac{\partial v}{\partial x}\right)^2+ \left(\frac{\partial v}{\partial y}\right)^2\right]\,dx\,dy \tag{8} \]

on the class \(\hat D(G)\) of functions for which it is defined and finite. First of all we establish that a function \(v(x,y)\in \hat D(G)\) has, in the sense of convergence in the mean, a boundary value \(F(P)\) with properties (7); and, conversely, a function \(F(P)\) given on \(\Gamma\) and possessing properties (7) can be extended to \(G\) in the weighted class \(\hat D(G)\). This is done according to the schemes indicated in [7], except that instead of Hardy’s inequality one uses the inequality

\[ \int_0^a \frac{dx}{x}\left|\int_0^x f(t)\,dt\right|^p \leq p^p \int_0^a x^{p-1}\ln^p\frac{M}{x}\,|f(x)|^p\,dx, \qquad p\geq 1,\ M\geq a . \]

Then the problem of finding a function \(v_0\) from \(\hat D(G)\), possessing the boundary value \(\varphi(P)\), on which the functional (8) attains its least value, is considered. Similarly to how this was done in Theorem 1, the existence of a unique function \(v_0\) with the indicated properties is proved, and, just as there, it is established that the function \(u_0=\ln\frac{M}{y}\cdot v_0\) is the unique solution of Problem B.

We note that in the case of the half-disk \(\{x^2+y^2<R^2,\ y>0\}\) the solution of Problem B can be written in explicit form, using the Green function of the operator \(L_1(u)\):

\[ \Gamma(x,y;\xi,\eta)=g(x,y;\xi,\eta)-\frac{R}{\rho}\,g(x,y;\xi^*,\eta^*), \]

\[ g(x,y;\xi,\eta)= \frac{1}{2\pi}\, \frac{1}{\ln\frac{M}{y}\,\ln\frac{M}{\eta}} \int_0^\pi \frac{d\alpha}{\left[(x-\xi)^2+y^2+\eta^2-2y\eta\cos\alpha\right]^{1/2}}, \]

\[ \frac{\xi^*}{\xi}=\frac{\eta^*}{\eta}=\frac{R^2}{\rho^2}, \qquad \rho^2=\xi^2+\eta^2 . \]

Then, by a limiting passage, one can obtain the solution of Problem B in the half-plane in the form

\[ u(x,y)=\frac{1}{2}\int_{-\infty}^{\infty} \frac{\varphi(t)\,dt}{\left[(x-t)^2+y^2\right]^{1/2}} . \tag{9} \]

It is easy to verify that for continuous functions \(\varphi(x)\) that decrease sufficiently rapidly at infinity, the function (9) satisfies the condition

\[ \lim_{y\to 0}\frac{u(x,y)}{\ln(1/y)}=\varphi(x). \]

The examples given lead to the following remark. It is known that the function (9) is the spatial potential of charges distributed on the axis \(Ox\) with density \(\varphi(x)\) (in this case one should

consider \(y\) as the distance to the axis \(Ox\). In exactly the same way, interpreting, for \(k>1\), equation (1) as the heat-propagation equation with thermal-conductivity coefficient \(y^\alpha\) \((k=1+\alpha)\), formula (6) may be interpreted as the stationary distribution of temperature in space, created by sources distributed along the axis \(Ox\) with density \(\varphi(x)\). Consequently, the formulations of boundary-value problems considered here make it possible to single out solutions of the corresponding equations that are singular on the boundary, and have a simple physical meaning.

Moscow Engineering Physics
Institute

Received
11 XI 1960

REFERENCES

¹ M. V. Keldysh, DAN, 77, No. 2 (1951).
² L. D. Kudryavtsev, Tr. Mat. Inst. im. V. A. Steklova AN SSSR, 55 (1959).
³ A. V. Bitsadze, Equations of Mixed Type, Moscow, 1959.
⁴ A. A. Vasharin, Izv. AN SSSR, ser. matem., 23, 421 (1959).
⁵ E. Magènes, Ann. mat. pura et appl., 48, 257 (1959).
⁶ P. I. Lizorkin, DAN, 126, No. 4 (1959).
⁷ P. I. Lizorkin, DAN, 134, No. 4 (1960).