{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:HDV4G7LVPKBMZHBQMB45JVABXQ","short_pith_number":"pith:HDV4G7LV","canonical_record":{"source":{"id":"2411.16048","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-25T02:28:58Z","cross_cats_sorted":[],"title_canon_sha256":"00ab1fbbc60076cb3730b023cfa924fe37f5af3bb88fcae980f79e33c37122be","abstract_canon_sha256":"17f3b428aae11877d3980fa6a43579decdac032a372e7f38e75dfe89203d5a35"},"schema_version":"1.0"},"canonical_sha256":"38ebc37d757a82cc9c306079d4d401bc0a56a0de99be11c2b03bb7d113852082","source":{"kind":"arxiv","id":"2411.16048","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.16048","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"arxiv_version","alias_value":"2411.16048v2","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.16048","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"pith_short_12","alias_value":"HDV4G7LVPKBM","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"pith_short_16","alias_value":"HDV4G7LVPKBMZHBQ","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"pith_short_8","alias_value":"HDV4G7LV","created_at":"2026-07-05T09:47:57Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:HDV4G7LVPKBMZHBQMB45JVABXQ","target":"record","payload":{"canonical_record":{"source":{"id":"2411.16048","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-25T02:28:58Z","cross_cats_sorted":[],"title_canon_sha256":"00ab1fbbc60076cb3730b023cfa924fe37f5af3bb88fcae980f79e33c37122be","abstract_canon_sha256":"17f3b428aae11877d3980fa6a43579decdac032a372e7f38e75dfe89203d5a35"},"schema_version":"1.0"},"canonical_sha256":"38ebc37d757a82cc9c306079d4d401bc0a56a0de99be11c2b03bb7d113852082","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:47:57.613748Z","signature_b64":"Q6OlDllqB6uUawYxgyNpOoBo/iOsH5h9C3JY2K8a8hDhWXrMpy2ZJAcnDKLNcDg26/vay4Dw8WMop4Vm73waBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38ebc37d757a82cc9c306079d4d401bc0a56a0de99be11c2b03bb7d113852082","last_reissued_at":"2026-07-05T09:47:57.613191Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:47:57.613191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2411.16048","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":"2026-07-05T09:47:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wKZ8/Z2SzAYuHwe7iPBHrDPWL4EPssBAU1iYwZQFCyQUr/hqhuRkGsSKN50hBYR2XFYNnykvp0zhy7K3nPgpDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T04:05:03.993494Z"},"content_sha256":"7bef9f8e07a3c43b25d5aa4b3bbb2587a19c382abc758b90ad6d470aed049e59","schema_version":"1.0","event_id":"sha256:7bef9f8e07a3c43b25d5aa4b3bbb2587a19c382abc758b90ad6d470aed049e59"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:HDV4G7LVPKBMZHBQMB45JVABXQ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Fine structure of rupture set for semilinear elliptic equation with singular nonlinearity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Wei Wang, Zhifei Zhang","submitted_at":"2024-11-25T02:28:58Z","abstract_excerpt":"In this paper, we study the stationary solutions of semilinear elliptic equation with singular nonlinearity $$ \\Delta u=u^{-p}+f,\\,\\,u\\geq 0\\text{ in }\\Omega\\subset\\mathbb{R}^n, $$ where $ n\\geq 2 $, $ p>1 $, $ \\Omega $ is a bounded domain, and $ f\\in L^q(\\Omega) $ with $ \\frac{1}{2}+\\frac{1}{2p}<\\frac{q}{n} $. We establish a sharp estimate for the Minkowski content of the rupture set $ \\{u=0\\} $ and demonstrate that this set is $ (n-2) $-rectifiable. For this, we examine the stratification of the rupture set based on the symmetry properties of tangent functions, leading to the proof of $ k $-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16048","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.16048/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T09:47:57Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AGKTQsexRc9jwzGVA9XtuWYwF/epwu4fSZwOGvP+pO8tqjqtK3YkKplM+psOeNo7/oGJzzw9uxMhusB65KT1Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-13T04:05:03.994472Z"},"content_sha256":"81f0c2203d38f5f9c191079f03b071c2670c227fe562dc3cd9ee31a246931603","schema_version":"1.0","event_id":"sha256:81f0c2203d38f5f9c191079f03b071c2670c227fe562dc3cd9ee31a246931603"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HDV4G7LVPKBMZHBQMB45JVABXQ/bundle.json","state_url":"https://pith.science/pith/HDV4G7LVPKBMZHBQMB45JVABXQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HDV4G7LVPKBMZHBQMB45JVABXQ/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":"2026-08-13T04:05:03Z","links":{"resolver":"https://pith.science/pith/HDV4G7LVPKBMZHBQMB45JVABXQ","bundle":"https://pith.science/pith/HDV4G7LVPKBMZHBQMB45JVABXQ/bundle.json","state":"https://pith.science/pith/HDV4G7LVPKBMZHBQMB45JVABXQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HDV4G7LVPKBMZHBQMB45JVABXQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:HDV4G7LVPKBMZHBQMB45JVABXQ","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":"17f3b428aae11877d3980fa6a43579decdac032a372e7f38e75dfe89203d5a35","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-25T02:28:58Z","title_canon_sha256":"00ab1fbbc60076cb3730b023cfa924fe37f5af3bb88fcae980f79e33c37122be"},"schema_version":"1.0","source":{"id":"2411.16048","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2411.16048","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"arxiv_version","alias_value":"2411.16048v2","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2411.16048","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"pith_short_12","alias_value":"HDV4G7LVPKBM","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"pith_short_16","alias_value":"HDV4G7LVPKBMZHBQ","created_at":"2026-07-05T09:47:57Z"},{"alias_kind":"pith_short_8","alias_value":"HDV4G7LV","created_at":"2026-07-05T09:47:57Z"}],"graph_snapshots":[{"event_id":"sha256:81f0c2203d38f5f9c191079f03b071c2670c227fe562dc3cd9ee31a246931603","target":"graph","created_at":"2026-07-05T09:47:57Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2411.16048/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we study the stationary solutions of semilinear elliptic equation with singular nonlinearity $$ \\Delta u=u^{-p}+f,\\,\\,u\\geq 0\\text{ in }\\Omega\\subset\\mathbb{R}^n, $$ where $ n\\geq 2 $, $ p>1 $, $ \\Omega $ is a bounded domain, and $ f\\in L^q(\\Omega) $ with $ \\frac{1}{2}+\\frac{1}{2p}<\\frac{q}{n} $. We establish a sharp estimate for the Minkowski content of the rupture set $ \\{u=0\\} $ and demonstrate that this set is $ (n-2) $-rectifiable. For this, we examine the stratification of the rupture set based on the symmetry properties of tangent functions, leading to the proof of $ k $-","authors_text":"Wei Wang, Zhifei Zhang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-25T02:28:58Z","title":"Fine structure of rupture set for semilinear elliptic equation with singular nonlinearity"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.16048","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":"sha256:7bef9f8e07a3c43b25d5aa4b3bbb2587a19c382abc758b90ad6d470aed049e59","target":"record","created_at":"2026-07-05T09:47:57Z","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":"17f3b428aae11877d3980fa6a43579decdac032a372e7f38e75dfe89203d5a35","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2024-11-25T02:28:58Z","title_canon_sha256":"00ab1fbbc60076cb3730b023cfa924fe37f5af3bb88fcae980f79e33c37122be"},"schema_version":"1.0","source":{"id":"2411.16048","kind":"arxiv","version":2}},"canonical_sha256":"38ebc37d757a82cc9c306079d4d401bc0a56a0de99be11c2b03bb7d113852082","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"38ebc37d757a82cc9c306079d4d401bc0a56a0de99be11c2b03bb7d113852082","first_computed_at":"2026-07-05T09:47:57.613191Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:47:57.613191Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"Q6OlDllqB6uUawYxgyNpOoBo/iOsH5h9C3JY2K8a8hDhWXrMpy2ZJAcnDKLNcDg26/vay4Dw8WMop4Vm73waBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:47:57.613748Z","signed_message":"canonical_sha256_bytes"},"source_id":"2411.16048","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7bef9f8e07a3c43b25d5aa4b3bbb2587a19c382abc758b90ad6d470aed049e59","sha256:81f0c2203d38f5f9c191079f03b071c2670c227fe562dc3cd9ee31a246931603"],"state_sha256":"568a27d44cda9b2655cee39a3f79ebe460ec92ae0ee235ea84d71732bd2e2da6"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"g3fKMVo9Dx9SmYSUB43c6q+57kllGIyq8j0oWmFGD4g0KIqp0IbTzk/4Vqp49OtBvFu0Ic1HFCJZ2pnZW7CPDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-13T04:05:04.001574Z","bundle_sha256":"916b4af6719186247fadba8d885050937a0def8d1d42d195df7b993d6925fb1f"}}