A regularity result for the p-laplacian near uniform ellipticity

الأصل · EN

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 Lq with q as large as it is desirable, provided p is sufficiently 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 it 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.