This article is cited in 23 papers

Solution of Dirichlet's problem for the equation $\Delta u=-1$ in a convex region

L. G. Makar-Limanov

M. V. Lomonosov Moscow State University

Abstract: Let $u$ be a solution of the following boundary-value problem: $u|_\Gamma=0$, where $\Gamma$ is a closed convex curve and $\Delta u=-1$ in the region $D$ bounded by $\Gamma$. Then $u$ has only one local maximum, and all its level curves are convex.

UDC: 517.9

Received: 06.10.1969

 English version: Mathematical Notes, 1971, 9:1, 52–53

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