A representation formula for the distributional normal derivative

TL;DR

This work develops an explicit integral representation formula for the distributional normal derivative of solutions to the Schrödinger-type problem $- \Delta u + V u = \mu$ in a smooth bounded domain with Dirichlet boundary data, where $V \in L^1_{\mathrm{loc}}(\Omega)$ is nonnegative and $\mu$ is a finite Borel measure. The authors introduce duality kernels $P_a$ solving $- \Delta P_a + V P_a = 0$ in $\Omega$ with $P_a = \delta_a$ on $\partial\Omega$ and use the precise boundary representative $\widehat{P_a}$ to show that, for a.e. $a \in \partial\Omega$, $\frac{\partial u}{\partial n}(a) = \int_\Omega \widehat{P_a} \, d\mu$. The representation is first proved for bounded $V$ and compactly supported data, then extended to general $V$ via truncations and measure-approximation, yielding a robust framework for boundary traces with measure data. As an application, the Hopf boundary point lemma is derived in an almost-everywhere sense when $V$ is a nonnegative Hopf potential, showcasing positivity of the boundary normal derivative under nontrivial nonnegative data and specific kernel positivity properties.

Abstract

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

Theorems & Definitions (18)

• Definition 1.1
• Theorem 1
• Definition 1.2
• Theorem 2
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof