Applications of Interpolation theory to the regularity of some equasilinear PDEs

TL;DR

This paper investigates the regularity of solutions to certain quasilinear PDEs using interpolation theory, providing new insights into the function spaces involved and extending existing results.

Contribution

It introduces novel regularity results for quasilinear PDE solutions and identifies new interpolation spaces relevant to these equations.

Findings

Regularity results for solutions in non-standard spaces

Identification of new interpolation spaces

Extension of existing PDE regularity theories

Abstract

We present some regularity results on the gradient of the weak or entropic-renormalized solution $u$ to the homogeneous Dirichlet problem for the quasilinear equations of the form \begin{equation*}\label{p-laplacian_eq} -{\rm div~}(|\nabla u|^{p-2}\nabla u)+V(x;u)=f, \end{equation*} where $\Omega$ is a bounded smooth domain of $\mathbb R^n$, $V$ is a nonlinear potential and $f$ belongs to non-standard spaces like Lorentz-Zygmund spaces. Moreover, we collect some well-known and new results for identifying some interpolation spaces and enrich some contents with details.