Bounded weak solutions with Orlicz space data: an overview

Abstract

It is well known 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 generalize this result by replacing the Laplacian with a degenerate elliptic operator, and we show that we can take the data $f$ in an Orlicz space $L^A(\Omega )$ that, in the classical case, lies strictly between $L^{\frac{n}{2}}(\Omega )$ and $L^q(\Omega )$, $q>\frac{n}{2}$.