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

### Werner Berndt

Universität Leipzig, Germany

## Abstract

The paper deals with the equation div $(ϱ(x)(▽u+f(x)))=0$ in $R_{N}$, where $ϱ∈L_{1}(R_{N})$ and $f∈L_{2}(R_{N},ϱ)$ are given smooth functions. The equation degenerates on the smooth surface $Γ={ϱ(x)=0}$ where $ϱ(x)$ behaves like a power of dist $(x,Γ)$. The following results are proved: 1. Existence and uniqueness (up to additive constants) of a solution with $∫∣▽u∣_{2}ϱdx<∞$; the proof uses a variational method in a weighted Sobolev space; 2. Regularity of the solution near $Γ$; 3. Convergence and correctness of a numerical (difference) method; 4. Convergence of an iteration method to solve the discrete problem.

## Cite this article

Werner Berndt, Solution of a degenerated elliptic equation of second order in an unbounded domain. Z. Anal. Anwend. 1 (1982), no. 3, pp. 53–68

DOI 10.4171/ZAA/19