The Hopf lemma for the Schrödinger operator

Abstract

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schrödinger operator $-\Delta +V$ with a nonnegative potential $V$ which merely belongs to $L_{\mathrm{loc}}^1(\Omega )$. More precisely, if $u\in W_0^{1,2}(\Omega )\cap L^2(\Omega ;V\mathrm{d}x)$ satisfies $-\Delta u+Vu=f$ on $\Omega$ for some nonnegative datum $f\in L^{\infty }(\Omega )$, $f\not\equiv 0$, then we show that at every point $a\in \partial \Omega$ 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 $$\{\begin{array}{l}\begin{array}{rlrl}-\Delta v+Vv& =0& & \text{in Ω,}\\ v& =\nu & & \text{on ∂Ω,}\end{array}\end{array}$$ involving the Dirac measure $\nu ={\delta }_a$ has a solution. More generally, we characterize the nonnegative finite Borel measures $\nu$ on $\partial \Omega$ for which the boundary value problem above has a solution in terms of the set where the Hopf lemma fails.

References in corpus (2)

• Hopf potentials for the Schrödinger operator
• On the nonexistence of Green's function and failure of the strong maximum principle