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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1609.08186","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-26T20:48:21Z","cross_cats_sorted":[],"title_canon_sha256":"3cfdaf5e87bd83d0edbb1b668c78b0720ac55d8cc6d0f81a608d3d9865690371","abstract_canon_sha256":"9a2df35c04dabc047788336e7230b2007b280596a93ce96a4c341c364353eed9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:13.757028Z","signature_b64":"uRWA3zN5Is3w+17xGYYpHMvG2VHebJAd7dIzqt4D8Xs5OGrgiy0QtuDPiYHX0WmUPRnLI1ODDyGXStunY9VQBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8a791057a0c2a2f93f97bc024161d62531124bb9b9a105931fbec792a1688582","last_reissued_at":"2026-05-18T00:02:13.756409Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:13.756409Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $W^{1,p}_0(\\Omega)$. 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