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 $\|u\|_{L^\infty(\Omega)} \leq C\|f\|_{L^q(\Omega)}$. 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 $L^q(\Omega)$, $q>\frac{n}2$.