Strong maximum principle for Schrödinger operators with singular potential
(2016) Annales de l’Institut Henri Poincaré - C - Non Linear Analysis — Vol. 33, n° 2, p. 477-493 (2016)
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- Abstract
- We prove that for every (p > 1) and for every potential (V in L^p), any nonnegative function satisfying (-Delta u + V u ge 0) in an open connected set of (R^N) is either identically zero or its level set ({u = 0}) has zero (W^{2, p}) capacity. This gives an affirmative answer to an open problem of Bénilan and Brezis concerning a bridge between Serrin-Stampacchia's strong maximum principle for (p > rac{N}{2}) and Ancona's strong maximum principle for (p = 1). The proof is based on the construction of suitable test functions depending on the level set ({u = 0}), and on the existence of solutions of the Dirichlet problem for the Schrödinger operator with diffuse measure data.
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Ponce, A., & Orsina, L. (2016). Strong maximum principle for Schrödinger operators with singular potential. Annales de l’Institut Henri Poincaré - C - Non Linear Analysis, 33(2), 477-493. https://doi.org/10.1016/j.anihpc.2014.11.004 (Original work published 2016)