A representation formula for the distributional normal derivative

TL;DR

This paper derives an integral formula for the distributional normal derivative of solutions to a Schrödinger-type PDE with measure data and demonstrates the almost everywhere validity of the Hopf lemma under nonnegative potential conditions.

Contribution

It provides a new integral representation for the distributional normal derivative in PDEs with measure data and extends the Hopf lemma to almost everywhere on the boundary for nonnegative potentials.

Findings

Established an integral representation formula for the distributional normal derivative.

Proved the Hopf lemma holds almost everywhere on the boundary under certain conditions.

Extended classical boundary behavior results to PDEs with measure data and nonnegative potentials.

Abstract

We prove an integral representation formula for the distributional normal derivative of solutions of $$ \left\{ \begin{aligned} - \Delta u + V u &= \mu && \text{in $\Omega$,}\\ u &= 0 && \text{on $\partial\Omega$,} \end{aligned} \right. $$ where $V \in L_{\mathrm{loc}}^1(\Omega)$ is a nonnegative function and $\mu$ is a finite Borel measure on $\Omega$. As an application, we show that the Hopf lemma holds almost everywhere on $\partial\Omega$ when $V$ is a nonnegative Hopf potential.