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arXiv · 2403.08575

Higher order Schauder estimates for degenerate or singular parabolic equations

Alessandro Audrito·Gabriele Fioravanti·Stefano Vita
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Abstract

In this paper, we complete the analysis initiated in [AFV24] establishing some higher order $C^{k+2,\alpha}$ Schauder estimates ($k \in \mathbb{N}$) for a a class of parabolic equations with weights that are degenerate/singular on a characteristic hyperplane. The $C^{2,\alpha}$-estimates are obtained through a blow-up argument and a Liouville theorem, while the higher order estimates are obtained by a fine iteration procedure. As a byproduct, we present two applications. First, we prove similar Schauder estimates when the degeneracy/singularity of the weight occurs on a regular hypersurface of cylindrical type. Second, we provide an alternative proof of the higher order boundary Harnack principles established in [BG16,Kuk22].

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Alessandro Audrito, Gabriele Fioravanti, Stefano Vita. 2024-03-13. Higher order Schauder estimates for degenerate or singular parabolic equations. https://doi.org/10.4171/rmi%2F1540

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