Optimal regularity for degenerate parabolic equations on a flat boundary

2025/04/07 by Hyungsung Yun, Yun, Hyungsung · 1 citation
Mathematics · #35K65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations #Primary 35B65 #Secondary 35D40

paper · pdf · doi:10.48550/arxiv.2504.04824

openalex publication_date 2025/04/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish the optimal regularity of viscosity solutions to u_t - x_n^γΔu = f, which arises in the regularity theory for the porous medium equation. Specifically, we prove that under the zero Dirichlet boundary condition on \x_n=0\, the optimal regularity of u up to the flat boundary \x_n=0\ is C^1,1-γ. Moreover, for the homogeneous equations, we establish that the optimal regularity of u is C^2,1-γ in the spatial variables, and that x_n^-γu is smooth in the variables x' and t.

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• On nondivergence form linear parabolic and elliptic equations with degenerate coefficients

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