## Differential and Integral Equations

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

### On a class of degenerate elliptic equations in weighted Hölder spaces

#### 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.