{"paper":{"title":"Degenerate elliptic operators: capacity, flux and separation","license":"","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adam Sikora, Derek W. Robinson","submitted_at":"2006-01-14T02:06:26Z","abstract_excerpt":"Let $S=\\{S_t\\}_{t\\geq0}$ be the semigroup generated on $L_2(\\Ri^d)$ by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients. Further let $\\Omega$ be an open subset of $\\Ri^d$ with Lipschitz continuous boundary $\\partial\\Omega$. We prove that $S$ leaves $L_2(\\Omega)$ invariant if, and only if, the capacity of the boundary with respect to $H$ is zero or if, and only if, the energy flux across the boundary is zero. The global result is based on an analogous local result."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0601351","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"}