Global Lorentz and Lorentz-Morrey estimates below the natural exponent for quasilinear equations

TL;DR

This paper establishes global Lorentz and Lorentz-Morrey gradient estimates for solutions to quasilinear equations modeled after the p-Laplacian, accommodating low integrability data and solutions with infinite energy.

Contribution

It provides new global estimates under mild geometric conditions and extends results to solutions with infinite energy and low integrability data.

Findings

Established Lorentz estimates for gradients of weak solutions.

Extended estimates to solutions with infinite energy.

Achieved boundary higher integrability results for homogeneous equations.

Abstract

Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the form $$\text{div}\,\mathcal{A}(x, \nabla u)=\text{div}\, |{\bf f}|^{p-2}{\bf f},$$ where $\text{div}\,\mathcal{A}(x, \nabla u)$ is modelled after the $p$-Laplacian, $p>1$. The estimates are global over bounded domains that satisfy a mild exterior uniform thickness condition that involves the $p$-capacity. The vector field datum ${\bf f}$ is allowed to have low degrees of integrability and thus solutions may not have finite $L^p$ energy. A higher integrability result at the boundary of the ground domain is also obtained for infinite energy solutions to the associated homogeneous equations.