Solution of a degenerated elliptic equation of second order in an unbounded domain

Abstract

The paper deals with the equation div $(\varrho (x)(▽u+f(x)))=0$ in ${\mathbb{R}}^N$, where $\varrho \in L_1({\mathbb{R}}^N)$ and $f\in L_2({\mathbb{R}}^N,\varrho )$ are given smooth functions. The equation degenerates on the smooth surface $\Gamma =\{\varrho (x)=0\}$ where $\varrho (x)$ behaves like a power of dist $(x,\Gamma )$. The following results are proved: 1. Existence and uniqueness (up to additive constants) of a solution with $\int |▽u{|}^2\varrho \mathrm{d}x<\infty$; the proof uses a variational method in a weighted Sobolev space; 2. Regularity of the solution near $\Gamma$; 3. Convergence and correctness of a numerical (difference) method; 4. Convergence of an iteration method to solve the discrete problem.