Differential and Integral Equations

On a class of degenerate elliptic equations in weighted Hölder spaces

Sergey I. Shmarev

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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.

Article information

Source
Differential Integral Equations Volume 17, Number 9-10 (2004), 1123-1148.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060316

Mathematical Reviews number (MathSciNet)
MR2082462

Subjects
Primary: 35J70: Degenerate elliptic equations
Secondary: 35B65: Smoothness and regularity of solutions 35J60: Nonlinear elliptic equations

Citation

Shmarev, Sergey I. On a class of degenerate elliptic equations in weighted Hölder spaces. Differential Integral Equations 17 (2004), no. 9-10, 1123--1148. https://projecteuclid.org/euclid.die/1356060316.


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