{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:SO72UKC5OYMIRVKVSP5OFVXUPP","short_pith_number":"pith:SO72UKC5","canonical_record":{"source":{"id":"1510.01596","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-06T14:40:25Z","cross_cats_sorted":[],"title_canon_sha256":"175ac5ac1bd7d4b35404c7cb2beb60555b4d482260df98bf4f85f2c206a02846","abstract_canon_sha256":"e42ea367c82f11c3b2ba8883c8bb42bfc2fc5ba232a84d0c461d64054678d1dd"},"schema_version":"1.0"},"canonical_sha256":"93bfaa285d761888d55593fae2d6f47be733a65beff3bac83ace05985fd9a4ff","source":{"kind":"arxiv","id":"1510.01596","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.01596","created_at":"2026-05-18T01:24:18Z"},{"alias_kind":"arxiv_version","alias_value":"1510.01596v2","created_at":"2026-05-18T01:24:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.01596","created_at":"2026-05-18T01:24:18Z"},{"alias_kind":"pith_short_12","alias_value":"SO72UKC5OYMI","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SO72UKC5OYMIRVKV","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SO72UKC5","created_at":"2026-05-18T12:29:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:SO72UKC5OYMIRVKVSP5OFVXUPP","target":"record","payload":{"canonical_record":{"source":{"id":"1510.01596","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-06T14:40:25Z","cross_cats_sorted":[],"title_canon_sha256":"175ac5ac1bd7d4b35404c7cb2beb60555b4d482260df98bf4f85f2c206a02846","abstract_canon_sha256":"e42ea367c82f11c3b2ba8883c8bb42bfc2fc5ba232a84d0c461d64054678d1dd"},"schema_version":"1.0"},"canonical_sha256":"93bfaa285d761888d55593fae2d6f47be733a65beff3bac83ace05985fd9a4ff","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:24:18.108573Z","signature_b64":"xEvpeX+YccYQBg0uk7BRl0q6A+5/MQDXAylNAhHjUZirNwCfCnLp45aTrTsxMuPbkDkT8vdGs3fYWNC28LyTAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"93bfaa285d761888d55593fae2d6f47be733a65beff3bac83ace05985fd9a4ff","last_reissued_at":"2026-05-18T01:24:18.108049Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:24:18.108049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1510.01596","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wub4ZM2DYv8E5NuAgSKdn3kpyL5Ec1AvBtI8CCXm6Hexpl+q5AQQjH9bDvKGGnmzsigMSH/aUeyydiiLPrIMCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T17:23:39.491435Z"},"content_sha256":"dcfaa5c7b07aac922fa33d1573286aa2700f802a9366eb27c3df7a22b74d9c05","schema_version":"1.0","event_id":"sha256:dcfaa5c7b07aac922fa33d1573286aa2700f802a9366eb27c3df7a22b74d9c05"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:SO72UKC5OYMIRVKVSP5OFVXUPP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Joaquim Serra, Xavier Cabre","submitted_at":"2015-10-06T14:40:25Z","abstract_excerpt":"We study nonlinear elliptic equations for operators corresponding to non-stable L\\'evy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) L\\'evy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen-Cahn type equations with a non-stable L\\'evy diffusion. These operators may still be realized as local operators using a system of PDEs ---in the spirit of the extension problem o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01596","kind":"arxiv","version":2},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:24:18Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4g5rUUoVicR2pDOIjWE+B6bOcpxSWZPbZNnbC42bEOjBLxULXMB2HH/1LI3Ku45P1HE8PLWPvgvZB5mTf5LNCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-06T17:23:39.492131Z"},"content_sha256":"a5fc681fdc94a1e3f80dbb59c318c7fd3f0d011be366760321b13ed34d2aca06","schema_version":"1.0","event_id":"sha256:a5fc681fdc94a1e3f80dbb59c318c7fd3f0d011be366760321b13ed34d2aca06"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/SO72UKC5OYMIRVKVSP5OFVXUPP/bundle.json","state_url":"https://pith.science/pith/SO72UKC5OYMIRVKVSP5OFVXUPP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/SO72UKC5OYMIRVKVSP5OFVXUPP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-06T17:23:39Z","links":{"resolver":"https://pith.science/pith/SO72UKC5OYMIRVKVSP5OFVXUPP","bundle":"https://pith.science/pith/SO72UKC5OYMIRVKVSP5OFVXUPP/bundle.json","state":"https://pith.science/pith/SO72UKC5OYMIRVKVSP5OFVXUPP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/SO72UKC5OYMIRVKVSP5OFVXUPP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:SO72UKC5OYMIRVKVSP5OFVXUPP","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":"e42ea367c82f11c3b2ba8883c8bb42bfc2fc5ba232a84d0c461d64054678d1dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-06T14:40:25Z","title_canon_sha256":"175ac5ac1bd7d4b35404c7cb2beb60555b4d482260df98bf4f85f2c206a02846"},"schema_version":"1.0","source":{"id":"1510.01596","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.01596","created_at":"2026-05-18T01:24:18Z"},{"alias_kind":"arxiv_version","alias_value":"1510.01596v2","created_at":"2026-05-18T01:24:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.01596","created_at":"2026-05-18T01:24:18Z"},{"alias_kind":"pith_short_12","alias_value":"SO72UKC5OYMI","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_16","alias_value":"SO72UKC5OYMIRVKV","created_at":"2026-05-18T12:29:42Z"},{"alias_kind":"pith_short_8","alias_value":"SO72UKC5","created_at":"2026-05-18T12:29:42Z"}],"graph_snapshots":[{"event_id":"sha256:a5fc681fdc94a1e3f80dbb59c318c7fd3f0d011be366760321b13ed34d2aca06","target":"graph","created_at":"2026-05-18T01:24:18Z","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 nonlinear elliptic equations for operators corresponding to non-stable L\\'evy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) L\\'evy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen-Cahn type equations with a non-stable L\\'evy diffusion. These operators may still be realized as local operators using a system of PDEs ---in the spirit of the extension problem o","authors_text":"Joaquim Serra, Xavier Cabre","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-06T14:40:25Z","title":"An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.01596","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:dcfaa5c7b07aac922fa33d1573286aa2700f802a9366eb27c3df7a22b74d9c05","target":"record","created_at":"2026-05-18T01:24:18Z","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":"e42ea367c82f11c3b2ba8883c8bb42bfc2fc5ba232a84d0c461d64054678d1dd","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-10-06T14:40:25Z","title_canon_sha256":"175ac5ac1bd7d4b35404c7cb2beb60555b4d482260df98bf4f85f2c206a02846"},"schema_version":"1.0","source":{"id":"1510.01596","kind":"arxiv","version":2}},"canonical_sha256":"93bfaa285d761888d55593fae2d6f47be733a65beff3bac83ace05985fd9a4ff","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"93bfaa285d761888d55593fae2d6f47be733a65beff3bac83ace05985fd9a4ff","first_computed_at":"2026-05-18T01:24:18.108049Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:24:18.108049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xEvpeX+YccYQBg0uk7BRl0q6A+5/MQDXAylNAhHjUZirNwCfCnLp45aTrTsxMuPbkDkT8vdGs3fYWNC28LyTAw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:24:18.108573Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.01596","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dcfaa5c7b07aac922fa33d1573286aa2700f802a9366eb27c3df7a22b74d9c05","sha256:a5fc681fdc94a1e3f80dbb59c318c7fd3f0d011be366760321b13ed34d2aca06"],"state_sha256":"f4561cd97f9262850f9e351919c7d402ad4d1c61678eef1cde94684169637b6b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JKrYBjBFAMPSvxdGswDl7m0E1ALfobspsGjO1h6MyT7FHlzKukpPkeHIJeaNKRxTMQMJPXAxCsr8IHK6+DTCDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-06T17:23:39.496212Z","bundle_sha256":"3db44ec60ff11995903b750d7d99c5fb9dbfcc185775a368f24f9e71fcc15e61"}}