$C^1$-regularity for degenerate diffusion equations

TL;DR

This paper proves that solutions to certain degenerate elliptic PDEs are continuously differentiable ($C^1$) under specific integrability conditions on the degeneracy law, using a novel recursive approximation method.

Contribution

It introduces a new recursive algorithm for renormalizing solutions, ensuring $C^1$-regularity under integrability conditions on the degeneracy law.

Findings

Solutions are proven to be $C^1$ under the given conditions.

A new recursive algorithm prevents degeneracy in the approximation process.

Universal estimates for the $C^1$-regularity of solutions are established.

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.