On the nonexistence of Green's function and failure of the strong maximum principle
(2020) Journal de mathématiques pures et appliquées — Vol. 134, p. 72-121 (2020)
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- Abstract
- (en) 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 \(- \Delta u + V u = \mu\) in \(\Omega\) involving a nonnegative finite Borel measure \(\mu\) has a distributional solution in \(W_{0}^{1, 1}(\Omega)\) if and only if \(\mu(Z) = 0\).
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Orsina, L., & Ponce, A. (2020). On the nonexistence of Green’s function and failure of the strong maximum principle. Journal de mathématiques pures et appliquées, 134, 72-121. https://doi.org/10.1016/j.matpur.2019.06.001 (Original work published 2020)