{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:IKAQASSZKZG5ADI3P4XPHQJJ3L","short_pith_number":"pith:IKAQASSZ","canonical_record":{"source":{"id":"1201.3486","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-17T11:30:54Z","cross_cats_sorted":[],"title_canon_sha256":"2fc17495e02ecaa0cb6fe373bc4e277e3b7867165a2f53f2f82b664f26becc8a","abstract_canon_sha256":"cb07d48402b6c152a9b5672be0cb21f1da69e72c957fcb248d9f59f7f2879a32"},"schema_version":"1.0"},"canonical_sha256":"4281004a59564dd00d1b7f2ef3c129dac4907117af05e257561fd85f46882580","source":{"kind":"arxiv","id":"1201.3486","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3486","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3486v1","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3486","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"IKAQASSZKZG5","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"IKAQASSZKZG5ADI3","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"IKAQASSZ","created_at":"2026-05-18T12:27:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:IKAQASSZKZG5ADI3P4XPHQJJ3L","target":"record","payload":{"canonical_record":{"source":{"id":"1201.3486","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-17T11:30:54Z","cross_cats_sorted":[],"title_canon_sha256":"2fc17495e02ecaa0cb6fe373bc4e277e3b7867165a2f53f2f82b664f26becc8a","abstract_canon_sha256":"cb07d48402b6c152a9b5672be0cb21f1da69e72c957fcb248d9f59f7f2879a32"},"schema_version":"1.0"},"canonical_sha256":"4281004a59564dd00d1b7f2ef3c129dac4907117af05e257561fd85f46882580","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:07.424820Z","signature_b64":"XZwlarkcQZsvkFZF/bOKONsNcoVLdfSY5qLJAaqLZnMVkL0gINirBDxYxVrfFHhvZAOWhxPantkbjRJEi/jOAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4281004a59564dd00d1b7f2ef3c129dac4907117af05e257561fd85f46882580","last_reissued_at":"2026-05-18T00:39:07.424234Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:07.424234Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.3486","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":"2026-05-18T00:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aTWffviTwcePUAa8KXYgsC0sVzK7JdojgsnDT+3vSgWitz25sWuTfLoq7fvjAwtNkmTDE1+gfqo8dyzuNV74AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:26:03.289686Z"},"content_sha256":"e52aa9cd9b9e00b991a4ee0ae099ed75d2e7923d07ffba4e02682a5f1b82eb62","schema_version":"1.0","event_id":"sha256:e52aa9cd9b9e00b991a4ee0ae099ed75d2e7923d07ffba4e02682a5f1b82eb62"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:IKAQASSZKZG5ADI3P4XPHQJJ3L","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniele Castorina, Manel Sanchon","submitted_at":"2012-01-17T11:30:54Z","abstract_excerpt":"In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \\textit{a priori} estimates for semi-stable solutions of $-\\Delta_p u= g(u)$ in a smooth bounded domain $\\Omega\\subset \\mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\\star$ when the domain is strictly convex. More precisely, we prove that $u^\\star\\in L^\\infty(\\Omega)$ if $n\\leq p+2$ and $u^\\star\\in L^{\\frac{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3486","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":"2026-05-18T00:39:07Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"G7wWrzX3IH+JjkX0Zqs99Y07oamoqv1m5KYw1Mwa7SgSdrCcXrkpJKv61ntGEhJr5JIPAhuVFCzhAz67pF6sBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:26:03.290283Z"},"content_sha256":"823a9b1ab249a935220a5f4e8f3a8ccc9f12bb9e3dd9c6dd523a7004da628fdc","schema_version":"1.0","event_id":"sha256:823a9b1ab249a935220a5f4e8f3a8ccc9f12bb9e3dd9c6dd523a7004da628fdc"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L/bundle.json","state_url":"https://pith.science/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L/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":"2026-06-08T18:26:03Z","links":{"resolver":"https://pith.science/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L","bundle":"https://pith.science/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L/bundle.json","state":"https://pith.science/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L/state.json","well_known_bundle":"https://pith.science/.well-known/pith/IKAQASSZKZG5ADI3P4XPHQJJ3L/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:IKAQASSZKZG5ADI3P4XPHQJJ3L","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":"cb07d48402b6c152a9b5672be0cb21f1da69e72c957fcb248d9f59f7f2879a32","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-17T11:30:54Z","title_canon_sha256":"2fc17495e02ecaa0cb6fe373bc4e277e3b7867165a2f53f2f82b664f26becc8a"},"schema_version":"1.0","source":{"id":"1201.3486","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.3486","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"arxiv_version","alias_value":"1201.3486v1","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.3486","created_at":"2026-05-18T00:39:07Z"},{"alias_kind":"pith_short_12","alias_value":"IKAQASSZKZG5","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_16","alias_value":"IKAQASSZKZG5ADI3","created_at":"2026-05-18T12:27:09Z"},{"alias_kind":"pith_short_8","alias_value":"IKAQASSZ","created_at":"2026-05-18T12:27:09Z"}],"graph_snapshots":[{"event_id":"sha256:823a9b1ab249a935220a5f4e8f3a8ccc9f12bb9e3dd9c6dd523a7004da628fdc","target":"graph","created_at":"2026-05-18T00:39:07Z","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":"In this paper we prove a Sobolev and a Morrey type inequality involving the mean curvature and the tangential gradient with respect to the level sets of the function that appears in the inequalities. Then, as an application, we establish \\textit{a priori} estimates for semi-stable solutions of $-\\Delta_p u= g(u)$ in a smooth bounded domain $\\Omega\\subset \\mathbb{R}^n$. In particular, we obtain new $L^r$ and $W^{1,r}$ bounds for the extremal solution $u^\\star$ when the domain is strictly convex. More precisely, we prove that $u^\\star\\in L^\\infty(\\Omega)$ if $n\\leq p+2$ and $u^\\star\\in L^{\\frac{","authors_text":"Daniele Castorina, Manel Sanchon","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-17T11:30:54Z","title":"Regularity of stable solutions of $p$-Laplace equations through geometric Sobolev type inequalities"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.3486","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:e52aa9cd9b9e00b991a4ee0ae099ed75d2e7923d07ffba4e02682a5f1b82eb62","target":"record","created_at":"2026-05-18T00:39:07Z","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":"cb07d48402b6c152a9b5672be0cb21f1da69e72c957fcb248d9f59f7f2879a32","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-01-17T11:30:54Z","title_canon_sha256":"2fc17495e02ecaa0cb6fe373bc4e277e3b7867165a2f53f2f82b664f26becc8a"},"schema_version":"1.0","source":{"id":"1201.3486","kind":"arxiv","version":1}},"canonical_sha256":"4281004a59564dd00d1b7f2ef3c129dac4907117af05e257561fd85f46882580","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4281004a59564dd00d1b7f2ef3c129dac4907117af05e257561fd85f46882580","first_computed_at":"2026-05-18T00:39:07.424234Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:07.424234Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XZwlarkcQZsvkFZF/bOKONsNcoVLdfSY5qLJAaqLZnMVkL0gINirBDxYxVrfFHhvZAOWhxPantkbjRJEi/jOAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:07.424820Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.3486","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e52aa9cd9b9e00b991a4ee0ae099ed75d2e7923d07ffba4e02682a5f1b82eb62","sha256:823a9b1ab249a935220a5f4e8f3a8ccc9f12bb9e3dd9c6dd523a7004da628fdc"],"state_sha256":"18d001c735e3d6b6458c7fd084e170abdd83e5c7a2832276e0860e5fec0ebf84"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"5EWE5rCNEcHP+FaP/cTNNFakU89bYPhc/fjZ5kxZDyifhCES2CEONiCk28CBlxj2tVHePVS8gE/EafyTwB13BQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:26:03.293972Z","bundle_sha256":"9f8cf408c231a97c6d7044a684724a066601bf544b4e49da05042f1b785cd40c"}}