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As a consequence, we obtain a new proof of the Moser-Trudinger inequality, of the Carleson-Chang result about the existence of extremals, and of the Struwe and Lamm-Robert-Struwe multiplicity result in the supercritical"},"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":"1608.07169","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-08-25T14:25:49Z","cross_cats_sorted":[],"title_canon_sha256":"566223f400da8a066337c9116d329f51565cd1c644be4796da7cb32b9ddd38ab","abstract_canon_sha256":"cb1365ca3185961fb7b87d2d579f08d75698fb9c3aea8fc478db352ca68d902f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:45:01.534599Z","signature_b64":"r256D1PUCO2AlKgbhNoT1qu9vepEUuwBRMuZ3oHA72CrSKc7/xDVTtOJ7jW1S3TP4DQmexd8+elr1OR5uU+qDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"000b6cb5ea2ef8e01ebdda89f4e5f6229c01ae66ecd2f05dfc328c933dec32c9","last_reissued_at":"2026-05-18T00:45:01.534228Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:45:01.534228Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Moser-Trudinger inequality and its extremals on a disk via energy estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gabriele Mancini, Luca Martinazzi","submitted_at":"2016-08-25T14:25:49Z","abstract_excerpt":"We study the Dirichlet energy of non-negative radially symmetric critical points $u_\\mu$ of the Moser-Trudinger inequality on the unit disc in $\\mathbb{R}^2$, and prove that it expands as $$4\\pi+\\frac{4\\pi}{\\mu^{4}}+o(\\mu^{-4})\\le \\int_{B_1}|\\nabla u_\\mu|^2dx\\le 4\\pi+\\frac{6\\pi}{\\mu^{4}}+o(\\mu^{-4}),\\quad \\text{as }\\mu\\to\\infty,$$ where $\\mu=u_\\mu(0)$ is the maximum of $u_\\mu$. 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