{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:K2KDBTWJHHSC5UQY5ROA72T22J","short_pith_number":"pith:K2KDBTWJ","canonical_record":{"source":{"id":"1208.4913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-24T07:10:46Z","cross_cats_sorted":[],"title_canon_sha256":"a8a26812334248dcf8c318f6ec142a089734d42d0c28aeba482ad72b380ee3a4","abstract_canon_sha256":"df5f333cf992b0f85a24a38df62b8be731875d33547be95c98306f0fbfb2fd83"},"schema_version":"1.0"},"canonical_sha256":"569430cec939e42ed218ec5c0fea7ad24b1f2efaa01b43dfb0fe0dd0d89401ae","source":{"kind":"arxiv","id":"1208.4913","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4913","created_at":"2026-05-18T02:25:20Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4913v1","created_at":"2026-05-18T02:25:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4913","created_at":"2026-05-18T02:25:20Z"},{"alias_kind":"pith_short_12","alias_value":"K2KDBTWJHHSC","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"K2KDBTWJHHSC5UQY","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"K2KDBTWJ","created_at":"2026-05-18T12:27:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:K2KDBTWJHHSC5UQY5ROA72T22J","target":"record","payload":{"canonical_record":{"source":{"id":"1208.4913","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-24T07:10:46Z","cross_cats_sorted":[],"title_canon_sha256":"a8a26812334248dcf8c318f6ec142a089734d42d0c28aeba482ad72b380ee3a4","abstract_canon_sha256":"df5f333cf992b0f85a24a38df62b8be731875d33547be95c98306f0fbfb2fd83"},"schema_version":"1.0"},"canonical_sha256":"569430cec939e42ed218ec5c0fea7ad24b1f2efaa01b43dfb0fe0dd0d89401ae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:25:20.018218Z","signature_b64":"8sfSsP5i7BOjERq8U55ezmkl6lYul/tri8odB87gfyuWZAwZtaBCkixsEw41wCfZEzRPSjPRhaqPOwPJN4LCAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"569430cec939e42ed218ec5c0fea7ad24b1f2efaa01b43dfb0fe0dd0d89401ae","last_reissued_at":"2026-05-18T02:25:20.017839Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:25:20.017839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.4913","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"if6cdoB3tDhvts34fdDbsQuzsx/PfrSuS31GJubqBonK6hXYWAk1SniNWrt8l+/1hPdD6Yod7WqPZzGn1+x8Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:18:41.740196Z"},"content_sha256":"e1a47e1cc74a24af43a76055a579ba2470a6991cf381b76dce6a8f455e7ea776","schema_version":"1.0","event_id":"sha256:e1a47e1cc74a24af43a76055a579ba2470a6991cf381b76dce6a8f455e7ea776"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:K2KDBTWJHHSC5UQY5ROA72T22J","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Anders Bj\\\"orn, Jana Bj\\\"orn","submitted_at":"2012-08-24T07:10:46Z","abstract_excerpt":"We study the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain Adams' criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Most of the results are new for open E (apart from those which are trivial in this case) and also on R^n."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4913","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T02:25:20Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/H1zIuamt3R8fRpK+uB3SdcgDXemVyfHJ//MeIelBNr7CWFDGAiDNvKP1kVrAOuATOsV5zuc1rwcTThwfmejAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T11:18:41.740883Z"},"content_sha256":"78dbbddf7d5bd5707bd61e1fedd362261e30098ac3ae0ffb4f60472492be1a7e","schema_version":"1.0","event_id":"sha256:78dbbddf7d5bd5707bd61e1fedd362261e30098ac3ae0ffb4f60472492be1a7e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K2KDBTWJHHSC5UQY5ROA72T22J/bundle.json","state_url":"https://pith.science/pith/K2KDBTWJHHSC5UQY5ROA72T22J/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K2KDBTWJHHSC5UQY5ROA72T22J/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":"2026-05-27T11:18:41Z","links":{"resolver":"https://pith.science/pith/K2KDBTWJHHSC5UQY5ROA72T22J","bundle":"https://pith.science/pith/K2KDBTWJHHSC5UQY5ROA72T22J/bundle.json","state":"https://pith.science/pith/K2KDBTWJHHSC5UQY5ROA72T22J/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K2KDBTWJHHSC5UQY5ROA72T22J/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:K2KDBTWJHHSC5UQY5ROA72T22J","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":"df5f333cf992b0f85a24a38df62b8be731875d33547be95c98306f0fbfb2fd83","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-24T07:10:46Z","title_canon_sha256":"a8a26812334248dcf8c318f6ec142a089734d42d0c28aeba482ad72b380ee3a4"},"schema_version":"1.0","source":{"id":"1208.4913","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4913","created_at":"2026-05-18T02:25:20Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4913v1","created_at":"2026-05-18T02:25:20Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4913","created_at":"2026-05-18T02:25:20Z"},{"alias_kind":"pith_short_12","alias_value":"K2KDBTWJHHSC","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"K2KDBTWJHHSC5UQY","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"K2KDBTWJ","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:78dbbddf7d5bd5707bd61e1fedd362261e30098ac3ae0ffb4f60472492be1a7e","target":"graph","created_at":"2026-05-18T02:25:20Z","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 the double obstacle problem for p-harmonic functions on arbitrary bounded nonopen sets E in quite general metric spaces. The Dirichlet and single obstacle problems are included as special cases. We obtain Adams' criterion for the solubility of the single obstacle problem and establish connections with fine potential theory. We also study when the minimal p-weak upper gradient of a function remains minimal when restricted to a nonopen subset. Most of the results are new for open E (apart from those which are trivial in this case) and also on R^n.","authors_text":"Anders Bj\\\"orn, Jana Bj\\\"orn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-24T07:10:46Z","title":"Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4913","kind":"arxiv","version":1},"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:e1a47e1cc74a24af43a76055a579ba2470a6991cf381b76dce6a8f455e7ea776","target":"record","created_at":"2026-05-18T02:25:20Z","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":"df5f333cf992b0f85a24a38df62b8be731875d33547be95c98306f0fbfb2fd83","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-08-24T07:10:46Z","title_canon_sha256":"a8a26812334248dcf8c318f6ec142a089734d42d0c28aeba482ad72b380ee3a4"},"schema_version":"1.0","source":{"id":"1208.4913","kind":"arxiv","version":1}},"canonical_sha256":"569430cec939e42ed218ec5c0fea7ad24b1f2efaa01b43dfb0fe0dd0d89401ae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"569430cec939e42ed218ec5c0fea7ad24b1f2efaa01b43dfb0fe0dd0d89401ae","first_computed_at":"2026-05-18T02:25:20.017839Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:25:20.017839Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"8sfSsP5i7BOjERq8U55ezmkl6lYul/tri8odB87gfyuWZAwZtaBCkixsEw41wCfZEzRPSjPRhaqPOwPJN4LCAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:25:20.018218Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.4913","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e1a47e1cc74a24af43a76055a579ba2470a6991cf381b76dce6a8f455e7ea776","sha256:78dbbddf7d5bd5707bd61e1fedd362261e30098ac3ae0ffb4f60472492be1a7e"],"state_sha256":"d430b797e5d1e509aca162fb37e91674996f7c2cf82d3a0081ad0ec1e0240044"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"lkIO810TRt2y56mGxW4IRwOpxd53Ts5mYfTLyLuWlChzeCQI/AcHku4GweZyyg+uszMBI25tOdtqj9p3haVkBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T11:18:41.744354Z","bundle_sha256":"2a27a68ceab5b71a8e4d30544db14509dd526d594d28cf6f3c5d3bd90a447c4b"}}