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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).


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.


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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.


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

zbMATH: 1150.35420
MathSciNet: MR2082462

Primary: 35J70
Secondary: 35B65, 35J60

Rights: Copyright © 2004 Khayyam Publishing, Inc.


Vol.17 • No. 9-10 • 2004
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