The Hopf lemma for the Schrödinger operator

TL;DR

The paper extends the Hopf boundary lemma to the Schrödinger operator $-\Delta + V$ with a nonnegative potential $V \in L_{\mathrm{loc}}^1(\Omega)$. It introduces the exceptional boundary set $\Sigma$ and a Schrödinger-adapted pointwise normal derivative, establishing that Hopf positivity holds precisely at boundary points not in $\Sigma$ and where the Dirac-boundary problem is solvable; when $V \in L^q(\Omega)$ with $q>N$, Hopf holds at all boundary points since $\Sigma=\emptyset$. A duality-solution framework for boundary measures is developed, linking solvability to whether the datum charges $\Sigma$, and a universal comparison principle clarifies how normal derivatives behave across solutions. These results quantify the boundary-interaction between the Green/Poisson kernel and the potential $V$, and identify sharp conditions under which Hopf’s positivity can fail or hold on the boundary. The analysis advances the understanding of boundary regularity for Schrödinger-type elliptic operators with rough potentials and has implications for boundary-value problems with measure data.

Abstract

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator $- Δ+ V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(Ω)$. More precisely, if $u \in W_0^{1, 2}(Ω) \cap L^2(Ω; V \mathrm{d}x)$ satisfies $- Δu + V u = f$ on $Ω$ for some nonnegative datum $f \in L^\infty(Ω)$, $f \not\equiv 0$, then we show that at every point $a \in \partialΩ$ where the classical normal derivative $\partial u(a) / \partial n$ exists and satisfies the Poisson representation formula, one has $\partial u(a) / \partial n > 0$ if and only if the boundary value problem $$ \begin{cases} \begin{aligned} - Δv + V v &= 0 && \text{in $Ω$,} \\ v &= ν&& \text{on $\partialΩ$,} \end{aligned} \end{cases} $$ involving the Dirac measure $ν= δ_a$ has a solution. More generally, we characterize the nonnegative finite Borel measures $ν$ on $\partialΩ$ for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

Theorems & Definitions (38)

• Theorem 1
• Definition 1.1
• Theorem 2
• Corollary 1.1
• Corollary 1.2
• Theorem 3
• Proposition 2.1
• Lemma 2.2
• Lemma 2.3
• proof : Proof of \ref{['prop:pointwise_normal_derivative']}