Higher differentiability for a class of problems under p,q subquadratic growth

TL;DR

This paper establishes higher differentiability results for solutions to nonlinear elliptic equations with p,q subquadratic growth, extending regularity theory to cases with Sobolev dependence on spatial variables.

Contribution

It provides new a-priori estimates ensuring local $W^{2,p}$ regularity for solutions under p,q growth conditions with Sobolev dependence on x.

Findings

Proves $W^{2,p}_{loc}$ regularity for solutions.

Extends regularity results to p,q growth with Sobolev dependence.

Provides a-priori estimates for nonlinear elliptic equations.

Abstract

We study the higher differentiability for nonlinear elliptic equation in divergence form $\mathcal{A}(x,Du)=b(x)$. The result covers the cases in which $\mathcal{A}(x, \xi)$ satisfies $p,q$ growth, with $1<p<2$ in $\xi$ and a Sobolev dependence of with respect to $x$. By means of an a-priori estimate we ensure the $W^{2,p}_{\mathrm{loc}}(\Omega)$-property for the solution of the boundary value problem.