Schauder estimates for parabolic equations with degenerate or singular weights

Abstract

We establish some \(C^{0,\alpha }\) and \(C^{1,\alpha }\) regularity estimates for a class of weighted parabolic problems in divergence form. The main novelty is that the weights may vanish or explode on a characteristic hyperplane \(\Sigma \) as a power \(a > -1\) of the distance to \(\Sigma \). The estimates we obtain are sharp with respect to the assumptions on coefficients and data. Our methods rely on a regularization of the equation and some uniform regularity estimates combined with a Liouville theorem and an approximation argument. As a corollary of our main result, we obtain similar \(C^{1,\alpha }\) estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.