On the nonexistence of Green's function and failure of the strong maximum principle

Abstract

Given any Borel function $V:\Omega \to [0,+\infty ]$ on a smooth bounded domain $\Omega \subset {\mathbb{R}}^N$, we establish that the strong maximum principle for the Schrödinger operator $-\Delta +V$ in $\Omega$ holds in each Sobolev-connected component of $\Omega \setminus Z$, where $Z\subset \Omega$ is the set of points which cannot carry a Green's function for $-\Delta +V$. More generally, we show that the equation $-\Delta u+Vu=\mu$ has a distributional solution in $W_0^{1,1}(\Omega )$ for a nonnegative finite Borel measure $\mu$ if and only if $\mu (Z)=0$.

References in corpus (4)

• Inverse Problem for a Curved Quantum Guide
• Hopf potentials for the Schrödinger operator
• Schroedinger operators involving singular potentials and measure data
• Critical weak-$L^p$ differentiability of singular integrals

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• A representation formula for the distributional normal derivative
• Duality arguments for linear elasticity problems with incompatible deformation fields
• An Agmon-Allegretto-Piepenbrink principle for Schroedinger operators