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.