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The equation describes the propagation of the time-harmonic electric field $\\Re\\{E(x)e^{i\\omega t}\\}$ in a nonlinear isotropic material $\\Omega$ with $\\lambda=-\\mu \\varepsilon \\omega^2\\leq 0$, where $\\mu$ and $\\varepsilon$ stand fo"},"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":"1609.03989","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-09-13T19:30:48Z","cross_cats_sorted":[],"title_canon_sha256":"349f9ba9f2954dff61d24751bc1cd692f685f79c676c86c24a17b5a0f1cebf36","abstract_canon_sha256":"8aa76565a61dbcb87aba00a431523002da8b510d0c290c8d72ebbb0df58f7c03"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:22.814015Z","signature_b64":"Z8BqqoIzRqZpb6+WRZqnLN/g6R9lehLylRXVBEiFY6LzzmKl/fo9RJ5Ra5zkHWZPFzLFV2pgS3faomuYqtF5DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2783418fe05478dd079a7825bb6d83491cab7dd7103cbce78b33d88a20521ed8","last_reissued_at":"2026-05-18T00:24:22.813555Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:22.813555Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Brezis-Nirenberg problem for the curl-curl operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jaros{\\l}aw Mederski","submitted_at":"2016-09-13T19:30:48Z","abstract_excerpt":"We look for solutions $E:\\Omega\\to\\mathbb{R}^3$ of the problem $$ \\left\\{ \\begin{aligned} &\\nabla\\times(\\nabla\\times E) +\\lambda E = |E|^{p-2}E &&\\quad \\text{in }\\Omega &\\nu\\times E = 0 &&\\quad \\text{on }\\partial\\Omega \\end{aligned} \\right. $$ on a bounded Lipschitz domain $\\Omega\\subset\\mathbb{R}^3$, where $\\nabla\\times$ denotes the curl operator in $\\mathbb{R}^3$. The equation describes the propagation of the time-harmonic electric field $\\Re\\{E(x)e^{i\\omega t}\\}$ in a nonlinear isotropic material $\\Omega$ with $\\lambda=-\\mu \\varepsilon \\omega^2\\leq 0$, where $\\mu$ and $\\varepsilon$ stand fo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.03989","kind":"arxiv","version":4},"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":"1609.03989","created_at":"2026-05-18T00:24:22.813617+00:00"},{"alias_kind":"arxiv_version","alias_value":"1609.03989v4","created_at":"2026-05-18T00:24:22.813617+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1609.03989","created_at":"2026-05-18T00:24:22.813617+00:00"},{"alias_kind":"pith_short_12","alias_value":"E6BUDD7AKR4N","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_16","alias_value":"E6BUDD7AKR4N2B42","created_at":"2026-05-18T12:30:12.583610+00:00"},{"alias_kind":"pith_short_8","alias_value":"E6BUDD7A","created_at":"2026-05-18T12:30:12.583610+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/E6BUDD7AKR4N2B42PAS3W3MDJE","json":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE.json","graph_json":"https://pith.science/api/pith-number/E6BUDD7AKR4N2B42PAS3W3MDJE/graph.json","events_json":"https://pith.science/api/pith-number/E6BUDD7AKR4N2B42PAS3W3MDJE/events.json","paper":"https://pith.science/paper/E6BUDD7A"},"agent_actions":{"view_html":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE","download_json":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE.json","view_paper":"https://pith.science/paper/E6BUDD7A","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1609.03989&json=true","fetch_graph":"https://pith.science/api/pith-number/E6BUDD7AKR4N2B42PAS3W3MDJE/graph.json","fetch_events":"https://pith.science/api/pith-number/E6BUDD7AKR4N2B42PAS3W3MDJE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE/action/storage_attestation","attest_author":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE/action/author_attestation","sign_citation":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE/action/citation_signature","submit_replication":"https://pith.science/pith/E6BUDD7AKR4N2B42PAS3W3MDJE/action/replication_record"}},"created_at":"2026-05-18T00:24:22.813617+00:00","updated_at":"2026-05-18T00:24:22.813617+00:00"}