The Hopf lemma for the Schr\"odinger operator

10 Jan 2020 Ponce Augusto C. Wilmet Nicolas

We prove the Hopf boundary point lemma for solutions of the Dirichlet problem involving the Schr\"odinger 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 $- \Delta u + V u = f$ 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 $$ \begin{cases} \begin{aligned} - \Delta v + V v &= 0 && \text{in $\Omega$,} \\ v &= \nu && \text{on $\partial\Omega$,} \end{aligned} \end{cases} $$ involving the Dirac measure $\nu = \delta_a$ has a solution... (read more)

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