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2004 On a class of degenerate elliptic equations in weighted Hölder spaces
Sergey I. Shmarev
Differential Integral Equations 17(9-10): 1123-1148 (2004).

Abstract

We study the Dirichlet problem for the degenerate elliptic equations \[ P_0\Delta\,u+\gamma(\nabla\,P_0,\nabla\,u)=f(x,u,\nabla\,u)\quad \mbox{in $\Omega$},\qquad \mbox{$u=0$ on $\partial \Omega$}, \] where $\gamma\geq 0$ is a given parameter, $\Omega\subset \mathbb{R}^n$ is an annular region, the given function $P_0(x)$ is such that $|\nabla\,P_0|+P_0\geq \epsilon> 0$ in $\overline\Omega$, and $P_0=0$ on the outer boundary of $\Omega$. The equation is degenerate elliptic when $\gamma>0$, while for $\gamma=0$ it transforms into the classical Poisson equation. We introduce the weighted Hölder spaces suitable for the study of the problem throughout the range of the parameter $\gamma\geq 0$. We derive the Schauder-type estimates and prove the existence of a unique classical solution. It is shown that in the case $\gamma>0$ the solution of the degenerate equation and the given function $P_0$ possess the same regularity properties. In the case $\gamma=0$ (the Poisson equation) the regularity of $u$ is better than the regularity of $P_0$. The proof is based on a new method of estimating the derivatives of solutions of the Poisson equation near the boundary of the problem domain which requires neither differentiation of the equation, nor straightening the boundary.

Citation

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Sergey I. Shmarev. "On a class of degenerate elliptic equations in weighted Hölder spaces." Differential Integral Equations 17 (9-10) 1123 - 1148, 2004.

Information

Published: 2004
First available in Project Euclid: 21 December 2012

zbMATH: 1150.35420
MathSciNet: MR2082462

Subjects:
Primary: 35J70
Secondary: 35B65, 35J60

Rights: Copyright © 2004 Khayyam Publishing, Inc.

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Vol.17 • No. 9-10 • 2004
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