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arXiv · 2208.11016

$C^1$-regularity for degenerate diffusion equations

Pêdra Andrade·Daniel Pellegrino·Edgard A. Pimentel·Eduardo V. Teixeira
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Abstract

We prove that any solution of a degenerate elliptic PDE is of class $C^1$, provided the inverse of the equation's degeneracy law satisfies an integrability criterium, viz. $\sigma^{-1} \in L^1\left (\frac{1}{\lambda} {\bf d}\lambda\right )$. The proof is based upon the construction of a sequence of converging tangent hyperplanes that approximate $u(x)$, near $x_0$, by an error of order $\text{o}(|x-x_0|)$. Explicit control of such hyperplanes is carried over through the construction, yielding universal estimates upon the ${C}^1$--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.

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Pêdra Andrade, Daniel Pellegrino, Edgard A. Pimentel, Eduardo V. Teixeira. 2022-08-23. $C^1$-regularity for degenerate diffusion equations. https://arxiv.org/abs/2208.11016

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