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.

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References in corpus (7)

• An extension problem related to the fractional Laplacian
• Regularity estimates for the solution and the free boundary to the obstacle problem for the fractional Laplacian
• Regularity Theory for Elliptic PDE
• Liouville type theorems and regularity of solutions to degenerate or singular problems part I: even solutions
• Liouville type theorems and regularity of solutions to degenerate or singular problems part II: odd solutions
• Higher order boundary Harnack principle via degenerate equations
• Schauder estimates for parabolic equations with degenerate or singular weights

Cited by in corpus (2)

• Schauder estimates for parabolic equations with degenerate or singular weights
• The Dirichlet problem on lower dimensional boundaries: Schauder estimates via perforated domains