Schauder estimates for parabolic equations with degenerate or singular weights

Abstract

AbstractWe establish some $$C^{0,\alpha }$$ C 0 , α and $$C^{1,\alpha }$$ C 1 , α 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 $$\Sigma $$ Σ as a power $$a > -1$$ a > - 1 of the distance to $$\Sigma $$ Σ . 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 $$C^{1,\alpha }$$ C 1 , α estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type.