{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:7CTAPYX7ULNT5S2SIRKIYUWJH2","short_pith_number":"pith:7CTAPYX7","canonical_record":{"source":{"id":"1807.10098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-26T12:49:28Z","cross_cats_sorted":[],"title_canon_sha256":"5a998fe2b135f6fbda22725742ee04d6ef1f5147791e928240f33910b9293b77","abstract_canon_sha256":"afdde650361917228d3f7a7bbdc0273eaeab26ec520766fbbb0dda40711c04a7"},"schema_version":"1.0"},"canonical_sha256":"f8a607e2ffa2db3ecb5244548c52c93eb321fa189eb3f75ee2ad75ba08779346","source":{"kind":"arxiv","id":"1807.10098","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.10098","created_at":"2026-05-18T00:09:45Z"},{"alias_kind":"arxiv_version","alias_value":"1807.10098v1","created_at":"2026-05-18T00:09:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10098","created_at":"2026-05-18T00:09:45Z"},{"alias_kind":"pith_short_12","alias_value":"7CTAPYX7ULNT","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7CTAPYX7ULNT5S2S","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7CTAPYX7","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:7CTAPYX7ULNT5S2SIRKIYUWJH2","target":"record","payload":{"canonical_record":{"source":{"id":"1807.10098","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-26T12:49:28Z","cross_cats_sorted":[],"title_canon_sha256":"5a998fe2b135f6fbda22725742ee04d6ef1f5147791e928240f33910b9293b77","abstract_canon_sha256":"afdde650361917228d3f7a7bbdc0273eaeab26ec520766fbbb0dda40711c04a7"},"schema_version":"1.0"},"canonical_sha256":"f8a607e2ffa2db3ecb5244548c52c93eb321fa189eb3f75ee2ad75ba08779346","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:45.425976Z","signature_b64":"ylCgYEqFZp6wyT2VzZm53NdT4PjlTM5CbAfzqWr5IsY0QxkYjU/V0wr8hgsxmzaVKnGuhtYULLnrcLEKfU2BCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8a607e2ffa2db3ecb5244548c52c93eb321fa189eb3f75ee2ad75ba08779346","last_reissued_at":"2026-05-18T00:09:45.425188Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:45.425188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.10098","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":"2026-05-18T00:09:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"F9QtMAo4b3BYj0E0452ETCG0tTktrcCr2Eoqow7bvmAoGz708NrtUPsfHTeIDFyZFOZ5E9j4nwP8s0pFO5H8Dw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:17:14.188278Z"},"content_sha256":"5664b5d91d253478160bb0f884834a4022f3246ea7ce86171b555a2d574a08a9","schema_version":"1.0","event_id":"sha256:5664b5d91d253478160bb0f884834a4022f3246ea7ce86171b555a2d574a08a9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:7CTAPYX7ULNT5S2SIRKIYUWJH2","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Glueing a peak to a non-zero limiting profile for a critical Moser-Trudinger equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gabriele Mancini, Pierre-Damien Thizy","submitted_at":"2018-07-26T12:49:28Z","abstract_excerpt":"Druet [6] proved that if $(f_\\gamma)_\\gamma$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_\\gamma)_\\gamma$ solves $$ \\begin{cases} &\\Delta u =f_\\gamma(x,u)\\,,~~ u>0\\text{ in }\\Omega\\,,\\\\ &u =0\\text{ on }\\partial\\Omega\\,, \\end{cases} $$ and converges weakly in $H^1_0$ to some $u_\\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\\ge 0$ such that the energy of $u_\\gamma$ converges to $4\\pi N$ plus the Dirichlet energy of $u_\\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10098","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":"2026-05-18T00:09:45Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dVKfAzcufD3F57v/5F+zx/4zUYw0EJfXSyAlePVFB2fpqLlJRqkS4BAQeZePSaLAxZ8ea4V89EMJCjc/5aINDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T01:17:14.188782Z"},"content_sha256":"d5a60a9e9aa106e0f21bd7838566be4bce8705ad5c76d4d8aebe5d8b25455feb","schema_version":"1.0","event_id":"sha256:d5a60a9e9aa106e0f21bd7838566be4bce8705ad5c76d4d8aebe5d8b25455feb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2/bundle.json","state_url":"https://pith.science/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2/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":"2026-06-02T01:17:14Z","links":{"resolver":"https://pith.science/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2","bundle":"https://pith.science/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2/bundle.json","state":"https://pith.science/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7CTAPYX7ULNT5S2SIRKIYUWJH2/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:7CTAPYX7ULNT5S2SIRKIYUWJH2","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":"afdde650361917228d3f7a7bbdc0273eaeab26ec520766fbbb0dda40711c04a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-26T12:49:28Z","title_canon_sha256":"5a998fe2b135f6fbda22725742ee04d6ef1f5147791e928240f33910b9293b77"},"schema_version":"1.0","source":{"id":"1807.10098","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.10098","created_at":"2026-05-18T00:09:45Z"},{"alias_kind":"arxiv_version","alias_value":"1807.10098v1","created_at":"2026-05-18T00:09:45Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.10098","created_at":"2026-05-18T00:09:45Z"},{"alias_kind":"pith_short_12","alias_value":"7CTAPYX7ULNT","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"7CTAPYX7ULNT5S2S","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"7CTAPYX7","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:d5a60a9e9aa106e0f21bd7838566be4bce8705ad5c76d4d8aebe5d8b25455feb","target":"graph","created_at":"2026-05-18T00:09:45Z","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":"Druet [6] proved that if $(f_\\gamma)_\\gamma$ is a sequence of Moser-Trudinger type nonlinearities with critical growth, and if $(u_\\gamma)_\\gamma$ solves $$ \\begin{cases} &\\Delta u =f_\\gamma(x,u)\\,,~~ u>0\\text{ in }\\Omega\\,,\\\\ &u =0\\text{ on }\\partial\\Omega\\,, \\end{cases} $$ and converges weakly in $H^1_0$ to some $u_\\infty$, then the Dirichlet energy is quantified, namely there exists an integer $N\\ge 0$ such that the energy of $u_\\gamma$ converges to $4\\pi N$ plus the Dirichlet energy of $u_\\infty$. As a crucial step to get the general existence results of [7], it was more recently proved in","authors_text":"Gabriele Mancini, Pierre-Damien Thizy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-26T12:49:28Z","title":"Glueing a peak to a non-zero limiting profile for a critical Moser-Trudinger equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.10098","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":"sha256:5664b5d91d253478160bb0f884834a4022f3246ea7ce86171b555a2d574a08a9","target":"record","created_at":"2026-05-18T00:09:45Z","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":"afdde650361917228d3f7a7bbdc0273eaeab26ec520766fbbb0dda40711c04a7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-26T12:49:28Z","title_canon_sha256":"5a998fe2b135f6fbda22725742ee04d6ef1f5147791e928240f33910b9293b77"},"schema_version":"1.0","source":{"id":"1807.10098","kind":"arxiv","version":1}},"canonical_sha256":"f8a607e2ffa2db3ecb5244548c52c93eb321fa189eb3f75ee2ad75ba08779346","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8a607e2ffa2db3ecb5244548c52c93eb321fa189eb3f75ee2ad75ba08779346","first_computed_at":"2026-05-18T00:09:45.425188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:09:45.425188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ylCgYEqFZp6wyT2VzZm53NdT4PjlTM5CbAfzqWr5IsY0QxkYjU/V0wr8hgsxmzaVKnGuhtYULLnrcLEKfU2BCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:09:45.425976Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.10098","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5664b5d91d253478160bb0f884834a4022f3246ea7ce86171b555a2d574a08a9","sha256:d5a60a9e9aa106e0f21bd7838566be4bce8705ad5c76d4d8aebe5d8b25455feb"],"state_sha256":"9696bc3c1fbfb0b8149f7f7bcf8a8c902775f4e63e9694dd81cf8c2ddc35edf4"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/AhQvgbHx4Rl3VhoUO166L3dsW0akkfhOkmji3Z8kgji0xDOvL6GQMFegjGYHhLBYaUFtWQ24+xusTM0JiUBBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T01:17:14.191701Z","bundle_sha256":"cbef38a123cfadc6f1d5a31662384dd9a20425cb18464b7e70a3dbd6b94c84da"}}