{"paper":{"title":"Stability for the Sobolev inequality: existence of a minimizer","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Tobias K\\\"onig","submitted_at":"2022-11-25T15:39:37Z","abstract_excerpt":"We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\\mathcal M$ and given by \\[ \\frac{\\|\\nabla f\\|_{L^2(\\mathbb R^d)}^2 - S_d \\|f\\|_{L^\\frac{2d}{d-2}(\\mathbb R^d)}^2}{ \\inf_{h \\in \\mathcal M} \\|\\nabla (f - h)\\|_{L^2(\\mathbb R^d)}^2 } \\geq c_{BE} > 0 \\qquad \\text{ for every } f \\in \\dot{H}^1(\\mathbb R^d),\\] which is due to Bianchi and Egnell, admits a minimizer for every $d \\geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $c_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2211.14185","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2211.14185/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}