{"paper":{"title":"Extremal functions in Poincare-Sobolev inequalities for functions of bounded variation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jean Van Schaftingen, Vincent Bouchez","submitted_at":"2010-01-26T11:07:29Z","abstract_excerpt":"If $\\Omega \\subset \\R^n$ is a smooth bounded domain and $q \\in (0, \\frac{n}{n-1})$ we consider the Poincare-Sobolev inequality \\[ c \\Bigl(\\int_{\\Omega} \\abs{u}^\\frac{n}{n-1}\\Bigr)^{1-\\frac{1}{n}} \\le \\int_{\\Omega} \\abs{Du}, \\] for every $u \\in \\mathrm{BV}(\\Omega)$ such that $\\int_{\\Omega} \\abs{u}^{q-1} u = 0$. We show that the sharp constant is achieved. We also consider the same inequality on an $n$--dimensional compact Riemannian manifold $M$. When $n \\ge 3$ and the scalar curvature is positive at some point, then the sharp constant is achieved. In the case $n \\ge 2$, we need the maximal sca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1001.4651","kind":"arxiv","version":2},"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"}