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We prove that this implies the weak upper semicontinuity of the functional $\\mathbb{D}(\\cdot)$ if and only if $p>\\frac{n}{n-1}$."},"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":"1906.06510","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2019-06-15T09:49:45Z","cross_cats_sorted":[],"title_canon_sha256":"1043011178870bf3c72ca1b4f0fce1878c61eddb9f2ae5aa4b67274fbf115885","abstract_canon_sha256":"9ec75ace8a587a1763916d035055b928087b9e33a70ff7280b58505ce5a85cbf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:43:13.492099Z","signature_b64":"pwab9D3i9333QoNZnuteJ3H7CR/uLDZlHtyj22KieT272LQavgbZKQn10+uoyl1kdncezLKoyqjvKd5mus1xAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56bbb8049c639fbe7dfb36f8fb5df50cea76ce8fb47159f7f4288271f4749dd5","last_reissued_at":"2026-05-17T23:43:13.491634Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:43:13.491634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the upper semicontinuity of a quasiconcave functional","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Denis Serre, Luigi De Rosa, Riccardo Tione","submitted_at":"2019-06-15T09:49:45Z","abstract_excerpt":"In the recent paper \\cite{SER}, the second author proved a divergence-quasiconcavity inequality for the following functional $ \\mathbb{D}(A)=\\int_{\\mathbb{T}^n} det(A(x))^{\\frac{1}{n-1}}\\,dx$ defined on the space of $p$-summable positive definite matrices with zero divergence. 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