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

Abstract

The paper deals with the equation div $(\varrho (x) (\bigtriangledown u + f(x))) = 0$ 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 $\int | \bigtriangledown u|^2 \varrho \mathrm d x< \infty$; 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.