$C^1$-regularity for degenerate diffusion equations

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. $σ^{-1} \in L^1\left (\frac{1}λ {\bf d}λ\right )$. 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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