MATHEMATICS

A. P. OSKOLKOV

ON THE SOLUTION OF BOUNDARY VALUE PROBLEMS FOR LINEAR ELLIPTIC EQUATIONS IN AN INFINITE DOMAIN

(Presented by Academician V. I. Smirnov on 31 V 1963)

1. Consider, in a three-dimensional domain \(\Omega\) lying outside a closed Lyapunov surface \(S\) of class \(L_2(1,\alpha)\), \(0<\alpha<1\), the linear uniformly elliptic equation

\[ Lu \equiv a_{ij}(x)\frac{\partial^2 u}{\partial x_i \partial x_j} + a_i(x)\frac{\partial u}{\partial x_i} + a(x)u=f(x), \qquad x=(x_1,x_2,x_3); \tag{1} \]

\[ \lambda_1 \sum_{i=1}^{3}\xi_i^2 \leq a_{ij}(x)\xi_i\xi_j \leq \lambda_2 \sum_{i=1}^{3}\xi_i^2, \qquad \lambda_1>0. \tag{2} \]

Let us introduce several definitions. We shall call a function \(\delta(t)\), \(0<a\leq t<\infty\), a Dini function if for it there exists and is finite the integral

\[ \int_a^\infty \frac{\delta(t)}{t}\,dt < \infty. \tag{3} \]

Let \(\delta(t)\) be a Dini function. Introduce the function

\[ I_\delta(\xi) \equiv \int_\xi^\infty \frac{\delta(t)}{t}\,dt, \qquad a\leq \xi<\infty. \tag{4} \]

Obviously, \(I_\delta(\xi)\to 0\) as \(\xi\to\infty\).

Denote by \(\Delta(\varepsilon)\), \(\varepsilon>0\), the class of Dini functions \(\delta(t)\), \(0<a\leq t<\infty\), satisfying the following conditions:

1) \(\delta(t)\) is a positive, monotonically decreasing, continuously differentiable Dini function;

2) \(|\delta'(t)|\leq \varepsilon \dfrac{\delta(t)}{t}\).

It is obvious that if \(\delta(t)\in\Delta(\varepsilon)\), then \(\delta(t)\geq t^{-\varepsilon}\). Further, if \(0<\varepsilon_1\leq\varepsilon_2\), then \(\Delta(\varepsilon_1)\supseteq\Delta(\varepsilon_2)\).

We shall assume that the origin of coordinates lies inside the surface \(S\), and let \(|\tilde{x}|\equiv \max\{|x|,|x'|\}\), \(x,x'\in\overline{\Omega}\). Suppose that the coefficients of equations (1)—(2) satisfy the following conditions:

\[ \|a_{ij}\|_{C^{(0,\alpha)}(\overline{\Omega})} \equiv \max_{x\in\overline{\Omega}}\sum_{i,j=1}^{3}|a_{ij}(x)| + \max_{x,x'\in\overline{\Omega}} \left\{ |\tilde{x}|^\alpha \sum_{i,j} \frac{|a_{ij}(x)-a_{ij}(x')|}{|x-x'|^\alpha} \right\} \leq K, \tag{5} \]

\[ \|a_i\|_{C^{(0,\alpha)}_1(\overline{\Omega})} \equiv \max \left\{ |x|\sum_{i=1}^{3}|a_i(x)| \right\} + \max \left\{ |\tilde{x}|^{1+\alpha} \sum_i \frac{|a_i(x)-a_i(x')|}{|x-x'|^\alpha} \right\} \leq K, \]

\[ a(x)\leq 0,\qquad \|a\|_{C^{(0,\alpha)}_{2,\delta/I_\delta}(\Delta,\overline{\Omega})} \equiv \]

\[ \equiv \max_{x\in\overline{\Omega}} \left\{ \frac{I_\delta(|x|)}{\delta(|x|)}\,|x|^2 |a(x)| \right\} + \max_{x,x'\in\overline{\Omega}} \left\{ \frac{I_\delta(|\tilde{x}|)}{\delta(|\tilde{x}|)} |\tilde{x}|^{2+\alpha} \frac{|a(x)-a(x')|}{|x-x'|^\alpha} \right\} \leq K. \tag{6} \]

We shall say that the free term \(f(x)\) belongs to the class \(C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\), \(0<\alpha<1\), \(\delta(t)\in\Delta(\varepsilon)\), if the following norm is defined and finite for it:

\[ \|f\|_{C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})} \equiv \max_{x\in\overline{\Omega}} \left\{ \frac{|x|^2}{\delta(|x|)}|f(x)| \right\} + \max_{x,x'\in\overline{\Omega}} \left\{ \frac{|\tilde{x}|^{2+\alpha}}{\delta(|\tilde{x}|)} \frac{|f(x)-f(x')|}{|x-x'|^\alpha} \right\} <\infty . \tag{7} \]

We shall further say that the function \(u(x)\) belongs to the class \(C_{\delta,l_\delta}^{(l,\alpha)}(\Delta,\overline{\Omega})\), \(l=0,1,2\); \(0<\alpha<1\); \(\delta(t)\in\Delta(\varepsilon)\), if the following norm is defined and finite for it:

\[ \begin{aligned} \|u\|_{C_{\delta,l_\delta}^{(l,\alpha)}(\Delta,\overline{\Omega})} &\equiv \max_{x\in\overline{\Omega}} \frac{|u(x)|}{l_\delta(|x|)} + \max_{x,x'\in\overline{\Omega}} \left\{ \frac{|\tilde{x}|^\alpha}{\delta(|\tilde{x}|)} \frac{|u(x)-u(x')|}{|x-x'|^\alpha} \right\} \\ &\quad+ \sum_{k=1}^{l} \max \left\{ \frac{|x|^k}{\delta(|x|)} \sum_{i,j=1}^{3}|D_{ij}^{k}u(x)| \right\} + \\ &\quad+ \sum_{k=1}^{l} \max \left\{ \frac{|\tilde{x}|^{k+\alpha}}{\delta(|\tilde{x}|)} \sum_{i,j} \frac{|D_{ij}^{k}u(x)-D_{ij}^{k}u(x')|}{|x-x'|^\alpha} \right\} <\infty . \end{aligned} \tag{8} \]

It is obvious that if the coefficients of the operator \(L\) satisfy conditions (5)—(6) and \(u(x)\in C_{\delta,l_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\), then \(Lu(x)\in C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\).

Let us consider, for equation (1)—(2), the Dirichlet problem with zero boundary condition at infinity:

\[ \begin{gathered} Lu=f(x)\quad \text{in } \Omega, \qquad f(x)\in C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega}),\\ u|_{S}=\varphi(x), \qquad \varphi(x)\in C^{(2,\alpha)}(S),\\ u(x)\to 0,\qquad |x|\to\infty . \end{gathered} \tag{9} \]

For solutions of problem (9) from the class \(C_{\delta,l_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\) the following a priori estimates of Schauder type are valid.

Theorem 1. If the coefficients of equation (1)—(2) satisfy conditions (5), (6), then for any solution of the boundary-value problem (9) from the class \(C_{\delta,l_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\) the following inequality is valid:

\[ \|u\|_{C_{\delta,l_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})} \le C(\lambda_1,K) \left[ \|f\|_{C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})} + \|\varphi\|_{C^{(2,\alpha)}(S)} + \max_{x\in\overline{\Omega}} \frac{|u(x)|}{l_\delta(|x|)} \right]. \tag{10} \]

Theorem 2. If the coefficients of equation (1)—(2) satisfy conditions (5), (6) and the condition

\[ \min_{x\in\overline{\Omega}} \left[ \sum_{i=1}^{3} a_{ii}(x) + a_i(x)x_i - (2+\varepsilon) \frac{a_{ij}(x)x_i x_j}{|x|^2} \right] \equiv K_1(\varepsilon)>0, \tag{11} \]

where \(\varepsilon>0\) is the same as in the definition of the class \(\Delta(\varepsilon)\), according to which the class of free terms \(C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\) is constructed, then for any solution of problem (9) from the class \(C_{\delta,l_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\) the inequality holds:

\[ \|u\|_{C_{\delta,l_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})} \le \frac{C(\lambda_1,K)}{K_1(\varepsilon)} \left[ \|f\|_{C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})} + \|\varphi\|_{C^{(2,\alpha)}(S)} \right]. \tag{12} \]

An example of an equation

\[ \Delta u=x^{-3},\quad |x|\ge 1; \qquad u(x)=\frac{\ln |x|}{|x|}\equiv l_\delta(|x|)\ln |x|, \tag{13} \]

shows that if \(K_1(\varepsilon)=0\), then the solution of equation (1)—(2), (5), (6) with right-hand side from the class \(C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\) may tend to zero as \(|x|\to\infty\) more slowly than \(I_\delta(|x|)\).

Let us note that condition (11) essentially imposes a restriction on the spread of the eigenvalues of the quadratic form \(a_{ij}(x)\xi_i\xi_j\).

Let \(0<\varepsilon<1\). Then condition (11) is also fulfilled for the Laplace operator (for it \(K_1(\varepsilon)=1-\varepsilon\)). It is easy to see that for the Laplace operator problem (9) is solvable in the class \(C_{\delta,I_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\), and then, using the a priori estimate (12), by the method of continuation with respect to a parameter we prove the theorem on solvability of the boundary-value problem (9) in the class \(C_{\delta,I_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\) also for equation (1)—(2).

Theorem 3. If the coefficients of equation (1)—(2) satisfy conditions (5), (6), (11), with \(0<\varepsilon<1\), then the boundary-value problem (9) has a unique solution in the class \(C_{\delta,I_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\).

Theorems 1—3 in the case \(\delta(t)=t^{-\varepsilon}\), \(0<\varepsilon<1\), were proved in the author’s paper \((^1)\).

Let now \(\varepsilon\geqslant 1\). Obviously, if \(0<\beta\leqslant\varepsilon\), then

\[ \Delta(\beta)\supseteq\Delta(\varepsilon),\quad K_1(\beta)\geqslant K_1(\varepsilon),\quad C_{2,\delta}^{(0,\alpha)}(\Delta(\beta),\overline{\Omega}) \supseteq C_{2,\delta}^{(0,\alpha)}(\Delta(\varepsilon),\overline{\Omega}), \]

\[ C_{\delta,I_\delta}^{(2,\alpha)}(\Delta(\beta),\overline{\Omega}) \supseteq C_{\delta,I_\delta}^{(2,\alpha)}(\Delta(\varepsilon),\overline{\Omega}). \]

Therefore the following theorem follows immediately from Theorem 3.

Theorem 4. If the coefficients of equation (1)—(2) satisfy conditions (5), (6), (11), with \(\varepsilon\geqslant 1\), then the boundary-value problem (9) has a unique solution in the class
\[ \bigcap_{0<\beta<1} C_{\delta,I_\delta}^{(2,\alpha)}(\Delta(\beta),\overline{\Omega}). \]

2. The solvability conditions for the boundary-value problem (9) in the class \(C_{\delta,I_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\) can be considerably broadened. Suppose that the coefficients of equation (1)—(2) satisfy conditions (5), (6), (11), \(0<\varepsilon<1\), \(f(x)\in C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\), and first consider the following spectral problem:

\[ Lu=f(x)+\lambda q(x)u \text{ in } \Omega,\qquad q(x)\in C_{2,\delta/I_\delta}^{(0,\alpha)}(\Delta,\overline{\Omega}), \tag{14} \]

\[ u|_S=0;\qquad u(x)\to 0,\ |x|\to\infty. \]

From Theorem 3 and the properties of \(q(x)\) it follows that the operator
\(\mathfrak{M}_0u\equiv L^{-1}(qu)\) is completely continuous in the space
\(C_{\delta,I_\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\). Therefore problem (14) reduces to the operator equation
\(u=F+\lambda_0\mathfrak{M}_0u\), \(F\equiv L^{-1}f\), in the space
\(C_{\delta,I_\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\) with the completely continuous operator \(\mathfrak{M}_0\). The spectrum of such an equation, and hence the spectrum of problem (14), is discrete and consists of at most a countable set of points in the complex \(\lambda\)-plane, with a possible accumulation point at infinity. If \(\lambda\) is not equal to any of these exceptional values, then the boundary-value problem (14) is solvable in the class \(C_{\delta,I_\delta}^{(2,\alpha)}(\Delta,\overline{\Omega})\) for any \(f(x)\in C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega})\).

We shall obtain a significant extension of Theorem 3 by considering the following spectral problem:

\[ Lu=f(x)+\mu A_i(x)\frac{\partial u}{\partial x_i}\text{ in } \Omega,\qquad A_i(x)\in C_1^{(0,\alpha)}(\overline{\Omega}), \tag{15} \]

\[ u|_S=0;\qquad u(x)\to 0,\ |x|\to\infty. \]

From Theorem 3 and the properties of the coefficients \(A_i(x)\) it follows that the operator

\[ \mathfrak{M}_1 u=L^{-1}\left(A_i\frac{\partial u}{\partial x_i}\right) \]

is completely continuous in the space \(C_{\delta, l/\delta}^{(1,\alpha)}(\Delta,\overline{\Omega})\). Therefore problem (15) reduces to the operator equation \(u=F+\mu\mathfrak{M}_1u\) in the space \(C_{\delta, l/\delta}^{(1,\alpha)}(\Delta,\overline{\Omega})\) with a completely continuous operator \(\mathfrak{M}_1\). Hence it follows that the spectrum of problem (15) is discrete and consists of no more than a countable set of points in the complex plane \((\mu)\), with a possible accumulation point at infinity.

Obviously, if

\[ \min_{x\in\overline{\Omega}} \left[ \sum_{i=1}^{3} a_{ii}(x) +(a_i-\mu A_i)x_i -(2+\varepsilon)\frac{a_{ij}x_i x_j}{|x|^2} \right]\equiv K_1(\varepsilon,\mu)>0, \tag{16} \]

then \(\mu\) is not a point of the spectrum of problem (15). The example of the equation

\[ \Delta u+\frac{l-2}{3}\,\frac{1}{x_i}\,\frac{\partial u}{\partial x_i} =|x|^{-(l+1)},\quad 1<l<2,\quad |x|\geqslant 1; \]

\[ u(x)=O\bigl(I_\delta(|x|)\ln |x|\bigr),\quad |x|\to\infty, \tag{17} \]

shows that if \(K_1(\varepsilon,\mu)=0\), then \(\mu\) can already be a point of the spectrum of problem (15).

1. Analogous results are valid for the boundary-value problem with normal derivative

\[ Lu=f(x)\ \text{in }\Omega,\qquad f(x)\in C_{2,\delta}^{(0,\alpha)}(\Delta,\overline{\Omega}), \]

\[ \left.\left(\frac{\partial u}{\partial n}+B(x)u\right)\right|_{S} =\psi(x),\qquad B(x)\leqslant 0,\qquad B(x),\psi(x)\in C^{(1,\alpha)}(S), \tag{18} \]

\[ u(x)\to 0,\qquad |x|\to\infty. \]

Leningrad Branch
of the V. A. Steklov Mathematical Institute
of the Academy of Sciences of the USSR

Received
24 V 1963

REFERENCES

1. A. P. Oskolkov, Vestn. LGU, No. 7, Ser. matem., mekh., astr., issue 2, 38 (1961).