{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:JNUBHTGGY7MWLDAUAOPSCKCYJ6","short_pith_number":"pith:JNUBHTGG","canonical_record":{"source":{"id":"1409.6284","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T19:14:44Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"471480470256cd5d5fce556c86053ea8591b76034c3beb32c3d1abfd52c974d2","abstract_canon_sha256":"91996e77129e1c0e7b21119e164a3b1917562a3d5c62d70e121ef9c8a2401ef8"},"schema_version":"1.0"},"canonical_sha256":"4b6813ccc6c7d9658c14039f2128584fb6c862d13539473fff3b8b1f40505784","source":{"kind":"arxiv","id":"1409.6284","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.6284","created_at":"2026-05-18T01:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1409.6284v2","created_at":"2026-05-18T01:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.6284","created_at":"2026-05-18T01:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"JNUBHTGGY7MW","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JNUBHTGGY7MWLDAU","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JNUBHTGG","created_at":"2026-05-18T12:28:35Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:JNUBHTGGY7MWLDAUAOPSCKCYJ6","target":"record","payload":{"canonical_record":{"source":{"id":"1409.6284","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T19:14:44Z","cross_cats_sorted":["math.FA"],"title_canon_sha256":"471480470256cd5d5fce556c86053ea8591b76034c3beb32c3d1abfd52c974d2","abstract_canon_sha256":"91996e77129e1c0e7b21119e164a3b1917562a3d5c62d70e121ef9c8a2401ef8"},"schema_version":"1.0"},"canonical_sha256":"4b6813ccc6c7d9658c14039f2128584fb6c862d13539473fff3b8b1f40505784","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:19:34.494550Z","signature_b64":"S+H7EcfaONbmeSEWmibd2aWuXsPfSmuyKlPNKO8u/qaw30uOzhkvyc6bizqyHjB3h/04USvKgzAkxORDQznMDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4b6813ccc6c7d9658c14039f2128584fb6c862d13539473fff3b8b1f40505784","last_reissued_at":"2026-05-18T01:19:34.494092Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:19:34.494092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1409.6284","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":"2026-05-18T01:19:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CFR3MKaux0+hY+22YZo8D2xkawnr46YknoYgzXDooSZ+ci9Y8v3hrT5RUt2Yg6CjXO5QwEFijl3N7rynGaQEDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T07:18:08.742647Z"},"content_sha256":"b381b124d89577c44f92824646044c2cd9b9cf9596fa84fbcee05d61fdfa6ccc","schema_version":"1.0","event_id":"sha256:b381b124d89577c44f92824646044c2cd9b9cf9596fa84fbcee05d61fdfa6ccc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:JNUBHTGGY7MWLDAUAOPSCKCYJ6","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The second eigenvalue of the fractional $p-$Laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Enea Parini, Lorenzo Brasco","submitted_at":"2014-09-22T19:14:44Z","abstract_excerpt":"We consider the eigenvalue problem for the {\\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\\Omega \\subset \\mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\\lambda_2(\\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \\[ \\inf "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6284","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":""},"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":"2026-05-18T01:19:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+uCdnzNIB6lvF7PYHwRM6CTY9LyR4qIu5pqZfF1OBxCxLA7K5rdoYivTIOSf+d4vTjBWlw6C4qW0If+tByMgAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-28T07:18:08.742984Z"},"content_sha256":"fa5cf56b47b294d336119e6e7367836c22d7f30d91a9da11e5da2da8f073fdab","schema_version":"1.0","event_id":"sha256:fa5cf56b47b294d336119e6e7367836c22d7f30d91a9da11e5da2da8f073fdab"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6/bundle.json","state_url":"https://pith.science/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6/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":"2026-06-28T07:18:08Z","links":{"resolver":"https://pith.science/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6","bundle":"https://pith.science/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6/bundle.json","state":"https://pith.science/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6/state.json","well_known_bundle":"https://pith.science/.well-known/pith/JNUBHTGGY7MWLDAUAOPSCKCYJ6/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:JNUBHTGGY7MWLDAUAOPSCKCYJ6","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":"91996e77129e1c0e7b21119e164a3b1917562a3d5c62d70e121ef9c8a2401ef8","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T19:14:44Z","title_canon_sha256":"471480470256cd5d5fce556c86053ea8591b76034c3beb32c3d1abfd52c974d2"},"schema_version":"1.0","source":{"id":"1409.6284","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1409.6284","created_at":"2026-05-18T01:19:34Z"},{"alias_kind":"arxiv_version","alias_value":"1409.6284v2","created_at":"2026-05-18T01:19:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1409.6284","created_at":"2026-05-18T01:19:34Z"},{"alias_kind":"pith_short_12","alias_value":"JNUBHTGGY7MW","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_16","alias_value":"JNUBHTGGY7MWLDAU","created_at":"2026-05-18T12:28:35Z"},{"alias_kind":"pith_short_8","alias_value":"JNUBHTGG","created_at":"2026-05-18T12:28:35Z"}],"graph_snapshots":[{"event_id":"sha256:fa5cf56b47b294d336119e6e7367836c22d7f30d91a9da11e5da2da8f073fdab","target":"graph","created_at":"2026-05-18T01:19:34Z","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 eigenvalue problem for the {\\it fractional $p-$Laplacian} in an open bounded, possibly disconnected set $\\Omega \\subset \\mathbb{R}^n$, under homogeneous Dirichlet boundary conditions. After discussing some regularity issues for eigenfuctions, we show that the second eigenvalue $\\lambda_2(\\Omega)$ is well-defined, and we characterize it by means of several equivalent variational formulations. In particular, we extend the mountain pass characterization of Cuesta, De Figueiredo and Gossez to the nonlocal and nonlinear setting. Finally, we consider the minimization problem \\[ \\inf ","authors_text":"Enea Parini, Lorenzo Brasco","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T19:14:44Z","title":"The second eigenvalue of the fractional $p-$Laplacian"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.6284","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:b381b124d89577c44f92824646044c2cd9b9cf9596fa84fbcee05d61fdfa6ccc","target":"record","created_at":"2026-05-18T01:19:34Z","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":"91996e77129e1c0e7b21119e164a3b1917562a3d5c62d70e121ef9c8a2401ef8","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-09-22T19:14:44Z","title_canon_sha256":"471480470256cd5d5fce556c86053ea8591b76034c3beb32c3d1abfd52c974d2"},"schema_version":"1.0","source":{"id":"1409.6284","kind":"arxiv","version":2}},"canonical_sha256":"4b6813ccc6c7d9658c14039f2128584fb6c862d13539473fff3b8b1f40505784","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4b6813ccc6c7d9658c14039f2128584fb6c862d13539473fff3b8b1f40505784","first_computed_at":"2026-05-18T01:19:34.494092Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:19:34.494092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S+H7EcfaONbmeSEWmibd2aWuXsPfSmuyKlPNKO8u/qaw30uOzhkvyc6bizqyHjB3h/04USvKgzAkxORDQznMDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:19:34.494550Z","signed_message":"canonical_sha256_bytes"},"source_id":"1409.6284","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b381b124d89577c44f92824646044c2cd9b9cf9596fa84fbcee05d61fdfa6ccc","sha256:fa5cf56b47b294d336119e6e7367836c22d7f30d91a9da11e5da2da8f073fdab"],"state_sha256":"1fa05b141d427f400285a8734908fadc0b951760da6219bd52285133af79bd77"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u93yc1io1ejFL3IQLhnQzEQVjgDLTpNcKeIvmNTemlaOIg3kcn+kYLyOuoUMrpJkVSdRT2kYQ1VxTRaLmVoOAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-28T07:18:08.744910Z","bundle_sha256":"f03fde287faa412976c53a7d0f1157f837c7e91246a00414b7146d4c731fd01d"}}