{"paper":{"title":"Extremal functions for Morrey's inequality in convex domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Erik Lindgren, Ryan Hynd","submitted_at":"2016-09-26T20:48:21Z","abstract_excerpt":"For a bounded domain $\\Omega\\subset \\mathbb{R}^n$ and $p>n$, Morrey's inequality implies that there is $c>0$ such that $$ c\\|u\\|^p_{\\infty}\\le \\int_\\Omega|Du|^pdx $$ for each $u$ belonging to the Sobolev space $W^{1,p}_0(\\Omega)$. We show that the ratio of any two extremal functions is constant provided that $\\Omega$ is convex. We also explain why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this property. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the $p$-Laplacia"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.08186","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"}