Regularity results for solutions to obstacle problems with Sobolev coeffcients

TL;DR

This paper proves higher differentiability of solutions to obstacle problems with Sobolev coefficients, extending regularity results under new assumptions that are independent of the dimension and improve previous bounds for certain cases.

Contribution

It introduces a novel Sobolev regularity assumption on the integrand's oscillation, enhancing regularity results for obstacle problems regardless of the spatial dimension.

Findings

Higher differentiability of solutions established

Regularity results independent of dimension n

Improved results for p ≤ n-2

Abstract

We establish the higher differentiability of solutions to a class of obstacle problems for integral functionals where the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned , under a suitable Sobolev assumption on the partial map that measures the oscillation of f with respect to the x variable. The main novelty is that such assumption is independent of the dimension n and that, in the case p<=n-2, improves previous known results.