Bounded weak solutions to elliptic PDE with data in Orlicz spaces

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 $\Vert u{\Vert }_{L^{\infty }(\Omega )}\le C\Vert f{\Vert }_{L^q(\Omega )}$. 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" $\Vert f{\Vert }_{L^A(\Omega )}/\Vert f{\Vert }_{L^{\frac{n}{2}}(\Omega )}$, generalizing a result due to Xu.