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

S. T. Huseynov   ; M. J. Aliyev

Vestnik Samarskogo universiteta. Estestvennonaučnaâ seriâ, Tome 30 (2024) no. 1, pp. 23-30

Résumé

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.

Détail

Keywords: uniform ellipticity, non-uniform degeneration spaces, fundamental solution.

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