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

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 $|| a_{ij}(x)||$ (function $\lambda_i(x)$) 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 $\Omega$.
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.