Strong maximum principle for Schrödinger operators with singular potential

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• 作者：Luigi Orsina^a ; ^{orsina@mat.uniroma1.it" class="auth_mail" title="E-mail the corresponding author} ; Augusto C. Ponce^b ; ^{Augusto.Ponce@uclouvain.be" class="auth_mail" title="E-mail the corresponding author}
• 关键词：primary ; 35B05 ; 35B50 ; secondary ; 31B15 ; 31B35
• 刊名：Annales de l'Institut Henri Poincare (C) Non Linear Analysis
• 出版年：2016
• 出版时间：March-April 2016
• 年：2016
• 卷：33
• 期：2
• 页码：477-493
• 全文大小：362 K

文摘

We prove that for every p>1$p>1$ and for every potential V∈L^p$V\in L^p$, any nonnegative function satisfying −Δu+Vu≥0$-\mathrm{\Delta }u+Vu\ge 0$ in an open connected set of R^N${\mathbb{R}}^N$ is either identically zero or its level set {u=0}$\{u=0\}$ has zero 33f10fb6ff9197f" title="Click to view the MathML source">W^2,p$W^{2,p}$ 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$p=1$. The proof is based on the construction of suitable test functions depending on the level set {u=0}$\{u=0\}$, and on the existence of solutions of the Dirichlet problem for the Schrödinger operator with diffuse measure data.