{"paper":{"title":"Ultracontractivity of Heat semigroups in $\\mathrm{L}^{2}\\left( \\Omega \\right)$ with non-local Robin boundary conditions using Nash's inequality","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Heat semigroups with generalized non-local Robin boundary conditions are ultracontractive on bounded Lipschitz domains.","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Christoph Schwerdt","submitted_at":"2026-05-13T12:09:11Z","abstract_excerpt":"We study heat equations $\\frac{\\partial u}{\\partial t} - \\operatorname{div} \\left( A \\nabla u \\right) = 0$ on bounded Lipschitz domains $\\Omega$ in $\\mathbb{R}^{d}$ for $d>2$, where $-\\operatorname{div} \\left( A \\nabla \\cdot \\right)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $\\nu \\cdot A \\nabla u + Bu = 0$ where $\\nu$ is the outer unit normal on $\\partial\\Omega$ and $B \\in \\mathcal{L} \\left( \\mathrm{L}^{2}\\left( \\partial \\Omega \\right) \\right)$ is a general operator which is allowed to destroy the p"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"Ultracontractivity of the solution semigroup is shown by using Nash's inequality on the Sobolev space H1(Omega) under quite mild assumptions on B.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The assumptions on the boundary operator B are 'quite mild' and the domain is bounded Lipschitz with uniform ellipticity of A; if these fail to hold in the stated generality, the Nash inequality application may not yield the ultracontractive bound.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Heat semigroups with non-local Robin boundary conditions on Lipschitz domains in R^d (d>2) are ultracontractive in L2 via Nash inequality on H1(Omega).","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Heat semigroups with generalized non-local Robin boundary conditions are ultracontractive on bounded Lipschitz domains.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a681581e0e4b5d0885e9bf04f0189fbac383deba1da72f41d2f15c47ca165994"},"source":{"id":"2605.13413","kind":"arxiv","version":1},"verdict":{"id":"45f86dad-92bd-4755-8169-85f23e9467c3","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:15:40.596983Z","strongest_claim":"Ultracontractivity of the solution semigroup is shown by using Nash's inequality on the Sobolev space H1(Omega) under quite mild assumptions on B.","one_line_summary":"Heat semigroups with non-local Robin boundary conditions on Lipschitz domains in R^d (d>2) are ultracontractive in L2 via Nash inequality on H1(Omega).","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The assumptions on the boundary operator B are 'quite mild' and the domain is bounded Lipschitz with uniform ellipticity of A; if these fail to hold in the stated generality, the Nash inequality application may not yield the ultracontractive bound.","pith_extraction_headline":"Heat semigroups with generalized non-local Robin boundary conditions are ultracontractive on bounded Lipschitz domains."},"references":{"count":30,"sample":[{"doi":"","year":null,"title":"Pitt , journal =","work_id":"4768fa2b-4ad4-4b92-9629-227947c1b07d","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Simader , journal =","work_id":"52d93f85-591f-4bd7-b517-20adf5a9355f","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Ben de Pagter , journal =","work_id":"28f00ff3-daec-40db-94ca-5ee67fa92433","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"Tosio Kato , journal =","work_id":"3304f50d-d8c5-4eb8-8f12-a48ddfe64b15","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"El-Maati Ouhabaz , publisher =","work_id":"68774ffd-fa31-4143-80c1-383091cc5a2b","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":30,"snapshot_sha256":"73fce88b74fe0b8500ee7a1a2bdd68f712d1c20cc803b0060a657d6b56d1c853","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"}