C¹-regularity for degenerate diffusion equations

2022/08/23 by Pêdra Andrade, Andrade, Pêdra, Daniel Pellegrino +5 · 4 citations
Computer Science · Engineering · Mathematics · #35B65 #35D40 #35J70 #37J60 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Numerical methods in inverse problems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.2208.11016

openalex publication_date 2022/08/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that any solution of a degenerate elliptic PDE is of class C¹, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. σ^-1 ∈ L¹ (\frac1λ \bf dλ ). The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate u(x), near x₀, by an error of order o(|x-x₀|). Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the C¹--regularity of solutions. Among the main new ingredients required in the proof, we develop an alternative recursive algorithm for the renormalization of approximating solutions. This new method is based on a technique tailored to prevent the sequence of degeneracy laws constructed through the process from being, itself, degenerate.

Cited by

• Regularity theory for degenerate fully nonlinear nonlocal equations with a Hamiltonian term
• Regularity for the normalized p-Laplacian equation with an arbitrary degeneracy law
• Boundary Hölder gradient estimates for fully nonlinear degenerate or singular parabolic equations
• Borderline regularity in singular free boundary problems
• Optimal regularity for degenerate elliptic equations with Hamiltonian terms

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