Regularity of solutions in semilinear elliptic theory

TL;DR

This paper investigates conditions on the nonlinear term in a semilinear Poisson equation that guarantee solutions are twice differentiable with Lipschitz continuous derivatives, with some conditions proven to be optimal.

Contribution

The paper establishes new regularity conditions on the nonlinear function ensuring solutions are in C^{1,1}, advancing understanding of solution smoothness in semilinear elliptic equations.

Findings

Solutions belong to C^{1,1} under certain conditions on f.

Some regularity conditions are proven to be sharp.

Provides a framework for analyzing solution regularity in semilinear elliptic PDEs.

Abstract

We study the semilinear Poisson equation \begin{equation} \label{pro} \Delta u = f(x, u) \hskip .2 in \text{in} \hskip .2 in B_1. \end{equation} Our main results provide conditions on $f$ which ensure that weak solutions of this equation belong to $C^{1,1}(B_{1/2})$. In some configurations, the conditions are sharp.