{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:LTQPAFUIK2CVLFVBAGAYJ7SKOS","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":"9c9a9c80e027c8249715ff1807f8215bb853580b4a41bc9fe622a2398b88cae5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-06T01:38:50Z","title_canon_sha256":"cf0162438142fa37c17d3e484aba0612029b08194c9451e8e1a27b77ffb54455"},"schema_version":"1.0","source":{"id":"1612.01623","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1612.01623","created_at":"2026-05-18T00:51:05Z"},{"alias_kind":"arxiv_version","alias_value":"1612.01623v2","created_at":"2026-05-18T00:51:05Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1612.01623","created_at":"2026-05-18T00:51:05Z"},{"alias_kind":"pith_short_12","alias_value":"LTQPAFUIK2CV","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_16","alias_value":"LTQPAFUIK2CVLFVB","created_at":"2026-05-18T12:30:29Z"},{"alias_kind":"pith_short_8","alias_value":"LTQPAFUI","created_at":"2026-05-18T12:30:29Z"}],"graph_snapshots":[{"event_id":"sha256:ded9e8b1e2049230cb2c1281b1db778da0ba437ce9df4be08c9be8c97cc40c64","target":"graph","created_at":"2026-05-18T00:51:05Z","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":"Using a direct approach, we prove a $2$-dimensional epiperimetric inequality for the one-phase problem in the scalar and vectorial cases and for the double-phase problem. From this we deduce, in dimension $2$, the $C^{1,\\alpha}$ regularity of the free-boundary in the scalar one-phase and double-phase problems, and of the reduced free-boundary in the vectorial case, without any restriction on the sign of the component functions. Furthermore we show that in the vectorial case the free boundary can end in a cusp.","authors_text":"Bozhidar Velichkov, Luca Spolaor","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-06T01:38:50Z","title":"An epiperimetric inequality for the regularity of some free boundary problems: the $2$-dimensional case"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1612.01623","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":"sha256:4166543208b6d95e0637de3e04fffac31a34cc83dba9811ea33e0bc597fb88d3","target":"record","created_at":"2026-05-18T00:51:05Z","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":"9c9a9c80e027c8249715ff1807f8215bb853580b4a41bc9fe622a2398b88cae5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2016-12-06T01:38:50Z","title_canon_sha256":"cf0162438142fa37c17d3e484aba0612029b08194c9451e8e1a27b77ffb54455"},"schema_version":"1.0","source":{"id":"1612.01623","kind":"arxiv","version":2}},"canonical_sha256":"5ce0f0168856855596a1018184fe4a749b3da44e9e955aa54e6bca6ce744a914","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5ce0f0168856855596a1018184fe4a749b3da44e9e955aa54e6bca6ce744a914","first_computed_at":"2026-05-18T00:51:05.566149Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:51:05.566149Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UnBnv6n9RnC1nb73sZanzCrxxpWaD2sd7kdEheig7OBXUnYNKp0t2LCZHSz0buYmgmmN7vDlooFZiPce5P7aDw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:51:05.566552Z","signed_message":"canonical_sha256_bytes"},"source_id":"1612.01623","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:4166543208b6d95e0637de3e04fffac31a34cc83dba9811ea33e0bc597fb88d3","sha256:ded9e8b1e2049230cb2c1281b1db778da0ba437ce9df4be08c9be8c97cc40c64"],"state_sha256":"612284c74505770634e2cf6b1d430795b3ade402739014e1effa04f7c23b152e"}