A regularity result for the p-laplacian near uniform ellipticity

We consider weak solutions to a class of Dirichlet boundary value problems invloving the $p$-Laplace operator, and prove that the second weak derivatives are in $L^{q}$ with $q$ as large as it is desirable, provided $p$ is sufficiently close to $p_0=2$. We show that this phenomenon is driven by the classical Calderón-Zygmund constant. As a byproduct of our analysis we show that $C^{1,α}$ regularity improves up to $C^{1,1^-}$, when p is close enough to 2. This result we believe it is particularly interesting in higher dimensions $n>2,$ when optimal $C^{1,α}$ regularity is related to the optimal regularity of $p$-harmonic mappings, which is still open.

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