Smoothness of the Solution of a Boundary Value Problem for Degenerate Elliptic Equations

This paper investigates boundary value problems for a class of elliptic equations exhibiting uniform and non-uniform degeneracy, including cases of non-monotonic degeneration. A key objective is to identify conditions on the coefficients under which solutions maintain ultimate smoothness, even in the presence of degeneracy. The analysis is grounded in several fundamental aspects of symmetry. Structural symmetry is reflected in the formulation of the differential operators; functional symmetry emerges in the properties of the associated weighted Sobolev spaces; and spectral symmetry plays a critical role in the behavior of the eigenvalues and eigenfunctions used to characterize solutions. By employing localization techniques, a priori estimates, and spectral theory, we establish new coefficient conditions ensuring smoothness in both semi-periodic and Dirichlet boundary settings. Moreover, we prove the boundedness and compactness of certain weighted operators, whose definitions and properties are tightly linked to underlying symmetries in the problem’s formulation. These results are not only of theoretical importance but also bear practical implications for numerical methods and models where symmetry principles influence solution regularity and operator behavior.