On the behavior of solutions of elliptic equations of second order in the neighborhood of a~singular boundary point

E. A. Mikheeva

Sbornik. Mathematics, Tome 9 (1969) no. 4, pp. 467-477

Résumé

The behavior of the solution of the linear elliptic equation \begin{equation} \label{1} \mathfrak Mu\equiv\sum_{i,\,k=1}^m a_{ik}(x)\frac{\partial^2u}{\partial x_i\partial x_k}+\sum_{i=1}^m b_i(x)\frac{\partial u}{\partial x_i}+c(x)u=0 \end{equation} with sufficiently smooth coefficients in a neighborhood of a singular boundary point is considered. Let $G$ be a bounded domain in $m$-space with boundary $\Gamma$. Let $x_0\in G$. For a nonnegative integer $n$ denote by $E_n$ the set of points in the complement of $G$ for which $$ 2^{-n}|x-x_0|\leqslant 2^{-(n-1)}. $$ The main result states that if the capacity $\gamma_n$ of the set $E_n$ satisfies the inequality $$ \gamma_n\leqslant\frac1{2^{n(k+m-2+\alpha)}}, $$ where $k$ is a nonnegative integer and $0\alpha1$, then the $k$th derivatives of the solution of (1) and the Hölder coefficients with exponents $\lambda\alpha$ of these derivatives are bounded constants which depend on $k$, $\alpha$, $\lambda$ and the constants of the elliptic equation and do not depend on the distance of $x_0$ from the boundary. Figure: 1. Bibliography: 7 titles.