Regularity results for weak solutions of elliptic PDEs below the natural exponent

TL;DR

This paper establishes new regularity estimates for weak solutions of elliptic PDEs in divergence form, specifically providing $L^p$ bounds for second derivatives when $p<2$, extending previous results.

Contribution

It generalizes existing regularity results to include $L^p$ estimates for second derivatives with $p<2$ for elliptic PDEs.

Findings

Established $L^p$ regularity estimates for second derivatives with $p<2$

Extended Miranda's results to a broader range of $p$

Provided new techniques for elliptic PDE regularity analysis

Abstract

We prove regularity estimates for weak solutions to the Dirichlet problem for a divergence form elliptic operator. We give $L^p$ estimates for the second derivative for $p<2$. Our work generalizes results due to Miranda [28].