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Here $\\lambda_1(\\Omega;D)$ is the first eigenvalue of the Laplace operator $-\\Delta$ with Dirichlet conditions on $\\partial\\Omega\\cap D$ and Neumann or Robin conditions on $\\partial\\Omega\\cap\\partial D$. The equivalent variational formulation $$\\lambda_1(\\Omega;D)=\\min\\left\\{\\int_\\Omega|\\nabla u|^2\\,dx+k\\int_{\\partial D}u^2\\,d\\mathcal{H}^{d-1}\\ :\\ u\\in H^1(D),\\ u=0\\hbox{ on }\\partial\\Omega\\cap D,\\ \\|u\\|"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1406.1627","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-06-06T10:08:56Z","cross_cats_sorted":[],"title_canon_sha256":"558ffb960b5408936c4a2b9f81c592414106dd4b1290862ebc5b0991741b05c6","abstract_canon_sha256":"9d0d958c2906ac71ed6d5c2cbf3e51c151a059c48e269bc1c1ad01249fba9d46"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:50:22.013497Z","signature_b64":"wdL4k421lwbk3oZ9JPc8qzkCO8AB9oiw2gYukhBssdF5/AzzKJCToU40mPx3G09mPq6/r+6WgF3buQEKB4xfDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c60a458d09653c9c742b5ab56ab18db4ce3ae56964f9459c7b01b428bfde617d","last_reissued_at":"2026-05-18T02:50:22.012812Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:50:22.012812Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The spectral drop problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Bozhidar Velichkov, Giuseppe Buttazzo","submitted_at":"2014-06-06T10:08:56Z","abstract_excerpt":"We consider spectral optimization problems of the form $$\\min\\Big\\{\\lambda_1(\\Omega;D):\\ \\Omega\\subset D,\\ |\\Omega|=1\\Big\\},$$ where $D$ is a given subset of the Euclidean space $\\mathbb{R}^d$. Here $\\lambda_1(\\Omega;D)$ is the first eigenvalue of the Laplace operator $-\\Delta$ with Dirichlet conditions on $\\partial\\Omega\\cap D$ and Neumann or Robin conditions on $\\partial\\Omega\\cap\\partial D$. The equivalent variational formulation $$\\lambda_1(\\Omega;D)=\\min\\left\\{\\int_\\Omega|\\nabla u|^2\\,dx+k\\int_{\\partial D}u^2\\,d\\mathcal{H}^{d-1}\\ :\\ u\\in H^1(D),\\ u=0\\hbox{ on }\\partial\\Omega\\cap D,\\ \\|u\\|"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.1627","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1406.1627","created_at":"2026-05-18T02:50:22.012891+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.1627v1","created_at":"2026-05-18T02:50:22.012891+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.1627","created_at":"2026-05-18T02:50:22.012891+00:00"},{"alias_kind":"pith_short_12","alias_value":"YYFELDIJMU6J","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_16","alias_value":"YYFELDIJMU6JY5BL","created_at":"2026-05-18T12:28:59.999130+00:00"},{"alias_kind":"pith_short_8","alias_value":"YYFELDIJ","created_at":"2026-05-18T12:28:59.999130+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT","json":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT.json","graph_json":"https://pith.science/api/pith-number/YYFELDIJMU6JY5BLLK2WVMMNWT/graph.json","events_json":"https://pith.science/api/pith-number/YYFELDIJMU6JY5BLLK2WVMMNWT/events.json","paper":"https://pith.science/paper/YYFELDIJ"},"agent_actions":{"view_html":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT","download_json":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT.json","view_paper":"https://pith.science/paper/YYFELDIJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.1627&json=true","fetch_graph":"https://pith.science/api/pith-number/YYFELDIJMU6JY5BLLK2WVMMNWT/graph.json","fetch_events":"https://pith.science/api/pith-number/YYFELDIJMU6JY5BLLK2WVMMNWT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT/action/storage_attestation","attest_author":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT/action/author_attestation","sign_citation":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT/action/citation_signature","submit_replication":"https://pith.science/pith/YYFELDIJMU6JY5BLLK2WVMMNWT/action/replication_record"}},"created_at":"2026-05-18T02:50:22.012891+00:00","updated_at":"2026-05-18T02:50:22.012891+00:00"}