A regularity result for the p-laplacian near uniform ellipticity

We consider weak solutions to a class of Dirichlet boundary value problems involving the p-Laplace operator, and prove that the second weak derivatives are in Lq with q as large as it is desirable, provided p is suffciently close to p0 = 2. We show that this phenomenon is driven by the classical Calderón-Zygmund constant. As a byproduct of our analysis we show that C1,α regularity improves up to C1,1, when p is close enough to 2. This result we believe is particularly interesting in higher dimensions n>2, when optimal C1,α regularity is related to the optimal regularity of p-harmonic mappings, which is still open.