Some auxiliary estimates for solutions to non-uniformly degenerate second-order elliptic equations

Abstract

We consider a class of second order elliptic equations in divergence form with non-uniform exponential degeneracy. The method used is based on the fact that the degeneracy rates of the eigenvalues of the matrix ||aij(x)|| (function λi(x)) are not the functions of unusual norm |x|, but of some anisotropic distance |x| a−. We assume that the Dirichlet problem for such equations is solvable in the classical sense for every continuous boundary function in any normal domain Ω. Estimates for the weak solutions of Dirichlet problem near the boundary point are obtained, and Green’s functions for second order non-uniformly degenerate elliptic equations are constructed.