A global fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data

TL;DR

This paper establishes a global fractional differentiability estimate for solutions to nonlinear elliptic equations with measure data, extending local regularity results to a weighted global setting using fractional Sobolev spaces.

Contribution

It introduces a global fractional Caccioppoli-type estimate for nonlinear elliptic problems with measure data, advancing the understanding of solution regularity in weighted fractional Sobolev spaces.

Findings

Proves a global fractional differentiability result for solutions.

Extends local fractional regularity to a weighted global context.

Provides a framework for analyzing solutions in weighted fractional Sobolev spaces.

Abstract

We prove a global fractional differentiability result via the fractional Caccioppoli-type estimate for solutions to nonlinear elliptic problems with measure data. This work is in fact inspired by the recent paper [B. Avelin, T. Kuusi, G. Mingione, {\em Nonlinear Calder\'on-Zygmund theory in the limiting case}, Arch. Rational. Mech. Anal. {\bf 227}(2018), 663--714], that was devoted to the local fractional regularity for the solutions to nonlinear elliptic equations with right-hand side measure, of type $-\mathrm{div}\, \mathcal{A}(\nabla u) = \mu$ in the limiting case. Being a contribution to recent results of identifying function classes that solutions to such problems could be defined, our aim in this work is to establish a global regularity result in a setting of weighted fractional Sobolev spaces, where the weights are powers of the distance function to the boundary of the smooth domains.