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In his 1992 paper, Suzuki proved that if $\\Omega$ is a simply-connected domain, then equation (1) admits a unique solution for $\\rho\\in[0,8\\pi)$. This result for $\\Omega$ a simply-connected domain has been extended to the case $\\rho=8\\pi$ by Chang, Chen and the second author. However, the uniqueness result for $\\Omega$ a multiply-connected domain has remained a long standing op"},"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":"1208.5228","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-26T15:22:54Z","cross_cats_sorted":[],"title_canon_sha256":"3d8cffcedac97f674eb827ec5e38e49aa5e9d4c058f4230fc9100b5cd074c26f","abstract_canon_sha256":"111f1299e1cf74d7bf578760c9c0fc8cf9466cb89536629cfd546d0aaaa3ac6e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:47:00.065383Z","signature_b64":"Ds8da+1iKQqKgQXq5LtqiuvGURBcMVfM54rK15CTpP/eHr/EJeN4mDqaUfibXc+hzZMWxuWK3Shel2dsRUGpCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6df7c7e1bf9140faf93ba2c07b50dfaa4c6f0fa15b389a77d894f04aa8ad0a48","last_reissued_at":"2026-05-18T03:47:00.064359Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:47:00.064359Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence and uniqueness for Mean Field Equations on multiply connected domains at the critical parameter","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Chang-shou Lin, Daniele Bartolucci","submitted_at":"2012-08-26T15:22:54Z","abstract_excerpt":"We consider the mean field equation: (1) \\Delta u+\\rho\\frac{e^u}{\\int_\\Omega e^u}=0 & \\hbox{in} \\;\\Omega, u=0 & \\hbox{on}\\;\\partial\\Omega, where $\\Omega\\subset \\mathbb{R}^2$ is an open and bounded domain of class $C^1$. In his 1992 paper, Suzuki proved that if $\\Omega$ is a simply-connected domain, then equation (1) admits a unique solution for $\\rho\\in[0,8\\pi)$. This result for $\\Omega$ a simply-connected domain has been extended to the case $\\rho=8\\pi$ by Chang, Chen and the second author. 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