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We prove that the topological boundary of a minimizer $\\Omega_k^*$ is composed of a relatively open regular part which is locally a graph of a $C^{1,\\alpha}$ function and a closed singular part, which is empty if $d<d^*$, contains at"},"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":"1609.01231","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-05T17:35:51Z","cross_cats_sorted":["math.FA","math.OC"],"title_canon_sha256":"c1c3b80b06089446c79e1ac4c84d0d3309d5a18c5782b25153c0a956eed96be4","abstract_canon_sha256":"0dd712f538fd4970ae4658ad207d00fae0d507847562466e9cc33bf7ee53db10"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:52:25.370026Z","signature_b64":"+MSn07P1tVRukHiQKFpy+iY7oJCOSoNMWLZspJpnFjpH0VJXag57pYngxIHuCjPF98K1yviSSBaO6/OCASswBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"80e70b9a93a3566f667a02d9883d200e5563d5d1c913ccba01f80f16724aac83","last_reissued_at":"2026-05-18T00:52:25.369442Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:52:25.369442Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Regularity of the optimal sets for some spectral functionals","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA","math.OC"],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Dario Mazzoleni, Susanna Terracini","submitted_at":"2016-09-05T17:35:51Z","abstract_excerpt":"In this paper we study the regularity of the optimal sets for the shape optimization problem \\[ \\min\\Big\\{\\lambda_1(\\Omega)+\\dots+\\lambda_k(\\Omega)\\ :\\ \\Omega\\subset\\mathbb{R}^d,\\ \\text{open}\\ ,\\ |\\Omega|=1\\Big\\}, \\] where $\\lambda_1(\\cdot),\\dots,\\lambda_k(\\cdot)$ denote the eigenvalues of the Dirichlet Laplacian and $|\\cdot|$ the $d$-dimensional Lebesgue measure. 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