{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:CTU44SWA5GI4W4XB5ZJZZZSIMM","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"2144689894f9f1c9e898888f9084e6ec83cdc9aca37fad82ed2096a64d51cbad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-09T17:15:12Z","title_canon_sha256":"2c56c3fde235a27ad9f551be025062e8255d7d7bc76b1fe03e9099efbf009d8f"},"schema_version":"1.0","source":{"id":"1104.1725","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1104.1725","created_at":"2026-05-18T04:07:06Z"},{"alias_kind":"arxiv_version","alias_value":"1104.1725v2","created_at":"2026-05-18T04:07:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1104.1725","created_at":"2026-05-18T04:07:06Z"},{"alias_kind":"pith_short_12","alias_value":"CTU44SWA5GI4","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"CTU44SWA5GI4W4XB","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"CTU44SWA","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:475500ded040ad3b681c0dd385eba12692ff3e44bd5b5631472fccc1c76eefa2","target":"graph","created_at":"2026-05-18T04:07:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study existence, unicity and other geometric properties of the minimizers of the energy functional $$ \\|u\\|^2_{H^s(\\Omega)}+\\int_\\Omega W(u)\\,dx, $$ where $\\|u\\|_{H^s(\\Omega)}$ denotes the total contribution from $\\Omega$ in the $H^s$ norm of $u$ and $W$ is a double-well potential. We also deal with the solutions of the related fractional elliptic Allen-Cahn equation on the entire space $\\mathbb{R}^n$.\n  The results collected here will also be useful for forthcoming papers, where the second and the third author will study the $\\Gamma$-convergence and the density estimates for level sets of ","authors_text":"Enrico Valdinoci, Giampiero Palatucci, Ovidiu Savin","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-09T17:15:12Z","title":"Local and global minimizers for a variational energy involving a fractional norm"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1104.1725","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:802c190cbd3bdaf9ed1a4a94272e0dd19cf61c03681b456232df322a09691295","target":"record","created_at":"2026-05-18T04:07:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"2144689894f9f1c9e898888f9084e6ec83cdc9aca37fad82ed2096a64d51cbad","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2011-04-09T17:15:12Z","title_canon_sha256":"2c56c3fde235a27ad9f551be025062e8255d7d7bc76b1fe03e9099efbf009d8f"},"schema_version":"1.0","source":{"id":"1104.1725","kind":"arxiv","version":2}},"canonical_sha256":"14e9ce4ac0e991cb72e1ee539ce648631560d5016c84021291fea77ddeadfb4a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14e9ce4ac0e991cb72e1ee539ce648631560d5016c84021291fea77ddeadfb4a","first_computed_at":"2026-05-18T04:07:06.688446Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:06.688446Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"MRO8YW6AWV5O+5grTQKsQOsqy4svneULdBkJUjgqi5bKLxVQ5qRhg9BTqMHu+cejicZ0G/EJJpGojjWVxaBjAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:06.688881Z","signed_message":"canonical_sha256_bytes"},"source_id":"1104.1725","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:802c190cbd3bdaf9ed1a4a94272e0dd19cf61c03681b456232df322a09691295","sha256:475500ded040ad3b681c0dd385eba12692ff3e44bd5b5631472fccc1c76eefa2"],"state_sha256":"5732fc8368abebda96c9f8f52710477b7e1b94f2e5fbb987c2460ef548999400"}