Bounded weak solutions to elliptic PDE with data in Orlicz spaces

TL;DR

This paper extends classical regularity results for elliptic PDEs by considering degenerate operators and data in Orlicz spaces, providing new bounds and generalizations involving entropy bump norms.

Contribution

It introduces bounds for solutions with data in Orlicz spaces for degenerate elliptic operators, generalizing previous results and involving entropy bump norms.

Findings

Solutions bounded in $L^ty$ norm with Orlicz space data

Extension to degenerate elliptic operators

Inclusion of entropy bump norm estimates

Abstract

A classical regularity result is that non-negative solutions to the Dirichlet problem $\Delta u =f$ in a bounded domain $\Omega$, where $f\in L^q(\Omega)$, $q>\frac{n}2$, satisfy $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. We extend this result in three ways: we replace the Laplacian with a degenerate elliptic operator; we show that we can take the data $f$ in an Orlicz space $L^A(\Omega)$ that lies strictly between $L^{\frac{n}{2}}(\Omega)$ and $L^q(\Omega)$, $q>\frac{n}2$; and we show that that we can replace the $L^A$ norm in the right-hand side by a smaller expression involving the logarithm of the "entropy bump" $\|f\|_{L^A(\Omega)}/\|f\|_{L^{\frac{n}{2}}(\Omega)}$, generalizing a result due to Xu.