Bounded weak solutions with Orlicz space data: an overview

TL;DR

This paper reviews the extension of classical regularity results for solutions of elliptic PDEs to cases where the data belongs to Orlicz spaces, broadening the scope beyond traditional Lebesgue spaces.

Contribution

It generalizes known bounds for solutions of elliptic equations to include data in Orlicz spaces, covering cases between $L^{n/2}$ and $L^q$ spaces.

Findings

Solutions are bounded when data is in certain Orlicz spaces.

The results extend classical $L^q$ bounds to more general function spaces.

Provides a framework for analyzing degenerate elliptic operators with Orlicz data.

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 $L^q(\Omega)$, $q>\frac{n}2$.