JournalszaaVol. 1, No. 3pp. 53–68

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

  • Werner Berndt

    Universität Leipzig, Germany
Solution of a degenerated elliptic equation of second order in an unbounded domain cover

Abstract

The paper deals with the equation div (ϱ(x)(u+f(x)))=0(\varrho (x) (\bigtriangledown u + f(x))) = 0 in RN\mathbb R^N, where ϱL1(RN)\varrho \in L_1 (\mathbb R^N) and fL2(RN,ϱ)f \in L_2 (\mathbb R^N, \varrho) are given smooth functions. The equation degenerates on the smooth surface Γ={ϱ(x)=0}\Gamma = \{\varrho (x) = 0\} where ϱ(x)\varrho (x) behaves like a power of dist (x,Γ)(x, \Gamma). The following results are proved: 1. Existence and uniqueness (up to additive constants) of a solution with u2ϱdx<\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 Γ\Gamma; 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