{"paper":{"title":"A Local Faber-Krahn inequality and Applications to Schr\\\"odinger's Equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","math.SP"],"primary_cat":"math.AP","authors_text":"Janna Lierl, Stefan Steinerberger","submitted_at":"2017-11-20T21:01:17Z","abstract_excerpt":"We prove a local Faber-Krahn inequality for solutions $u$ to the Dirichlet problem for $\\Delta + V$ on an arbitrary domain $\\Omega$ in $\\mathbb{R}^n$. Suppose a solution $u$ assumes a global maximum at some point $x_0 \\in \\Omega$ and $u(x_0)>0$. Let $T(x_0)$ be the smallest time at which a Brownian motion, started at $x_0$, has exited the domain $\\Omega$ with probability $\\ge 1/2$. For nice (e.g., convex) domains, $T(x_0) \\asymp d(x_0,\\partial\\Omega)^2$ but we make no assumption on the geometry of the domain. Our main result is that there exists a ball $B$ of radius $\\asymp T(x_0)^{1/2}$ such "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1711.07541","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}