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This article is cited in 1 scientific paper (total in 1 paper)
On the behavior of solutions of elliptic equations of second order in the neighborhood of a singular boundary point
E. A. Mikheeva
Abstract:
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 $0<\alpha<1$, 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.
Received: 04.09.1968
Citation:
E. A. Mikheeva, “On the behavior of solutions of elliptic equations of second order in the neighborhood of a singular boundary point”, Mat. Sb. (N.S.), 80(122):4(12) (1969), 503–512; Math. USSR-Sb., 9:4 (1969), 467–477
Linking options:
https://www.mathnet.ru/eng/sm3639https://doi.org/10.1070/SM1969v009n04ABEH002056 https://www.mathnet.ru/eng/sm/v122/i4/p503
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Abstract page: | 322 | Russian version PDF: | 73 | English version PDF: | 6 | References: | 25 |
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