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

Abstract

Lorentz and Lorentz-Morrey estimates are obtained for gradients of very weak solutions to quasilinear equations of the form $$\text{div}\mathcal{A}(x,\nabla u)=\text{div}|\mathbf{f}{|}^{p-2}\mathbf{f},$$ 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 $\mathbf{f}$ 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.

37 pages

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