Strong maximum principle for Schrödinger operators with singular potential

Orsina, Luigi; Ponce, Augusto C.

doi:10.1016/j.anihpc.2014.11.004

Orsina, Luigi; Ponce, Augusto C.

Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 2, pp. 477-493.

Abstract

We prove that for every $p > 1$ and for every potential $V \in L^{p}$ , any nonnegative function satisfying $- Δ u + V u \geq 0$ 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 $p > \frac{N}{2}$ 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.

MR Zbl

DOI: 10.1016/j.anihpc.2014.11.004

Classification: 35B05, 35B50, 31B15, 31B35
Keywords: Maximum principle, Schrödinger operator, Kato's inequality, Capacity

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     author = {Orsina, Luigi and Ponce, Augusto C.},
     title = {Strong maximum principle for {Schr\"odinger} operators with singular potential},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {477--493},
     publisher = {Elsevier},
     volume = {33},
     number = {2},
     year = {2016},
     doi = {10.1016/j.anihpc.2014.11.004},
     zbl = {1342.35083},
     mrnumber = {3465383},
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     url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2014.11.004/}
}

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Orsina, Luigi; Ponce, Augusto C. Strong maximum principle for Schrödinger operators with singular potential. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 2, pp. 477-493. doi : 10.1016/j.anihpc.2014.11.004. http://archive.numdam.org/articles/10.1016/j.anihpc.2014.11.004/

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