{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:TMIKIG2H3ELX5DYKN56TGBJ72C","short_pith_number":"pith:TMIKIG2H","canonical_record":{"source":{"id":"1211.2364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-11T01:25:16Z","cross_cats_sorted":[],"title_canon_sha256":"e6ae6be5ba0226ceb476ac38ec7da9e025c39bbf33081867b8653e8195e2b46f","abstract_canon_sha256":"d6620621efa234226e236f06522cd70158661e1cacff4db089e2287d9b8ace94"},"schema_version":"1.0"},"canonical_sha256":"9b10a41b47d9177e8f0a6f7d33053fd0bbd5d3ac678cad60729f430a10249160","source":{"kind":"arxiv","id":"1211.2364","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2364","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2364v1","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2364","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"pith_short_12","alias_value":"TMIKIG2H3ELX","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TMIKIG2H3ELX5DYK","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TMIKIG2H","created_at":"2026-05-18T12:27:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:TMIKIG2H3ELX5DYKN56TGBJ72C","target":"record","payload":{"canonical_record":{"source":{"id":"1211.2364","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-11T01:25:16Z","cross_cats_sorted":[],"title_canon_sha256":"e6ae6be5ba0226ceb476ac38ec7da9e025c39bbf33081867b8653e8195e2b46f","abstract_canon_sha256":"d6620621efa234226e236f06522cd70158661e1cacff4db089e2287d9b8ace94"},"schema_version":"1.0"},"canonical_sha256":"9b10a41b47d9177e8f0a6f7d33053fd0bbd5d3ac678cad60729f430a10249160","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:57:53.189356Z","signature_b64":"UKaHirG9krlYKcbnKw7NljDJB/zgBnsC3fMtJFjnZgO1wQev2Hgbko9EAwhpQ8ggLbe/rKdFpsJ/biZz3MUUCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9b10a41b47d9177e8f0a6f7d33053fd0bbd5d3ac678cad60729f430a10249160","last_reissued_at":"2026-05-18T02:57:53.188710Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:57:53.188710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1211.2364","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":"2026-05-18T02:57:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"nx7aF2UIrgT9oki7cmMIY4FlLGPf7o3kA4l2hOWiHWDs7+WlsK5+NC7zc0WtDtCvywYYqP3TMcMx274YITDsDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:33:12.847181Z"},"content_sha256":"566c789953d533dd4703dafb3883b9993e4bedc72a6d72b6ccc694654996d781","schema_version":"1.0","event_id":"sha256:566c789953d533dd4703dafb3883b9993e4bedc72a6d72b6ccc694654996d781"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:TMIKIG2H3ELX5DYKN56TGBJ72C","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Boundary clustered layers near the higher critical exponents","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Angela Pistoia, M\\'onica Clapp, Nils Ackermann","submitted_at":"2012-11-11T01:25:16Z","abstract_excerpt":"We consider the supercritical problem {equation*} -\\Delta u=|u| ^{p-2}u\\text{\\in}\\Omega,\\quad u=0\\text{\\on}\\partial\\Omega, {equation*} where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\\ast}:=\\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\\mathbb{R}^{N-k})$ in $L^{q}(\\mathbb{R}^{N-k})$, $1\\leq k\\leq N-3.$ We show that in some suitable domains $\\Omega$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\\partial\\Omega$ a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2364","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":"2026-05-18T02:57:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xrwT5OvrTVs/jgWskKAcOAmCfSFLVL+ZkQJOIsbNVj+2pEEXatpCQeOsgWIOjtaz6om+y8yv1F6qf3s5yHa5DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T16:33:12.847547Z"},"content_sha256":"ae7d726c945bc06f9c742314539c8fb0331383c4ffe79dea31f87f7d26c82a60","schema_version":"1.0","event_id":"sha256:ae7d726c945bc06f9c742314539c8fb0331383c4ffe79dea31f87f7d26c82a60"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TMIKIG2H3ELX5DYKN56TGBJ72C/bundle.json","state_url":"https://pith.science/pith/TMIKIG2H3ELX5DYKN56TGBJ72C/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TMIKIG2H3ELX5DYKN56TGBJ72C/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":"2026-06-02T16:33:12Z","links":{"resolver":"https://pith.science/pith/TMIKIG2H3ELX5DYKN56TGBJ72C","bundle":"https://pith.science/pith/TMIKIG2H3ELX5DYKN56TGBJ72C/bundle.json","state":"https://pith.science/pith/TMIKIG2H3ELX5DYKN56TGBJ72C/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TMIKIG2H3ELX5DYKN56TGBJ72C/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:TMIKIG2H3ELX5DYKN56TGBJ72C","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":"d6620621efa234226e236f06522cd70158661e1cacff4db089e2287d9b8ace94","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-11T01:25:16Z","title_canon_sha256":"e6ae6be5ba0226ceb476ac38ec7da9e025c39bbf33081867b8653e8195e2b46f"},"schema_version":"1.0","source":{"id":"1211.2364","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1211.2364","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"arxiv_version","alias_value":"1211.2364v1","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1211.2364","created_at":"2026-05-18T02:57:53Z"},{"alias_kind":"pith_short_12","alias_value":"TMIKIG2H3ELX","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_16","alias_value":"TMIKIG2H3ELX5DYK","created_at":"2026-05-18T12:27:23Z"},{"alias_kind":"pith_short_8","alias_value":"TMIKIG2H","created_at":"2026-05-18T12:27:23Z"}],"graph_snapshots":[{"event_id":"sha256:ae7d726c945bc06f9c742314539c8fb0331383c4ffe79dea31f87f7d26c82a60","target":"graph","created_at":"2026-05-18T02:57:53Z","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 consider the supercritical problem {equation*} -\\Delta u=|u| ^{p-2}u\\text{\\in}\\Omega,\\quad u=0\\text{\\on}\\partial\\Omega, {equation*} where $\\Omega$ is a bounded smooth domain in $\\mathbb{R}^{N}$ and $p$ smaller than the critical exponent $2_{N,k}^{\\ast}:=\\frac{2(N-k)}{N-k-2}$ for the Sobolev embedding of $H^{1}(\\mathbb{R}^{N-k})$ in $L^{q}(\\mathbb{R}^{N-k})$, $1\\leq k\\leq N-3.$ We show that in some suitable domains $\\Omega$ there are positive and sign changing solutions with positive and negative layers which concentrate along one or several $k$-dimensional submanifolds of $\\partial\\Omega$ a","authors_text":"Angela Pistoia, M\\'onica Clapp, Nils Ackermann","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-11T01:25:16Z","title":"Boundary clustered layers near the higher critical exponents"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1211.2364","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":"sha256:566c789953d533dd4703dafb3883b9993e4bedc72a6d72b6ccc694654996d781","target":"record","created_at":"2026-05-18T02:57:53Z","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":"d6620621efa234226e236f06522cd70158661e1cacff4db089e2287d9b8ace94","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-11-11T01:25:16Z","title_canon_sha256":"e6ae6be5ba0226ceb476ac38ec7da9e025c39bbf33081867b8653e8195e2b46f"},"schema_version":"1.0","source":{"id":"1211.2364","kind":"arxiv","version":1}},"canonical_sha256":"9b10a41b47d9177e8f0a6f7d33053fd0bbd5d3ac678cad60729f430a10249160","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9b10a41b47d9177e8f0a6f7d33053fd0bbd5d3ac678cad60729f430a10249160","first_computed_at":"2026-05-18T02:57:53.188710Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:57:53.188710Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UKaHirG9krlYKcbnKw7NljDJB/zgBnsC3fMtJFjnZgO1wQev2Hgbko9EAwhpQ8ggLbe/rKdFpsJ/biZz3MUUCg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:57:53.189356Z","signed_message":"canonical_sha256_bytes"},"source_id":"1211.2364","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:566c789953d533dd4703dafb3883b9993e4bedc72a6d72b6ccc694654996d781","sha256:ae7d726c945bc06f9c742314539c8fb0331383c4ffe79dea31f87f7d26c82a60"],"state_sha256":"a81ed5a6d3e8b5dddad013b1c8fa040696de39098e0fcb2f63e0ca2ace6f4e83"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DECzlP2OWUgDBQqDGAFrL0s557JBNdmELOqgNA4mFxvdbWL68q35T/CbON7XOOFlns719g8shZqB6iWKu20XAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T16:33:12.849527Z","bundle_sha256":"90abc56993bdac59046ef812c5ab452e22a16b9b68522dca22094ff75ae0f684"}}