{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:6YKQ3NXFK3X5ICPFPH2G4CXGYA","short_pith_number":"pith:6YKQ3NXF","canonical_record":{"source":{"id":"1805.11549","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-29T15:56:06Z","cross_cats_sorted":[],"title_canon_sha256":"d66387fdc1951ec6bb0bd91493cc0c07e742a512fd296fddc60a47c8b739a5f5","abstract_canon_sha256":"1411fdc084bf594cd3b488f07fb31b46edb43c2f1f430cfc0f855925e2759620"},"schema_version":"1.0"},"canonical_sha256":"f6150db6e556efd409e579f46e0ae6c025ac9caf121545efbf2c741422bc1272","source":{"kind":"arxiv","id":"1805.11549","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.11549","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"arxiv_version","alias_value":"1805.11549v2","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.11549","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"pith_short_12","alias_value":"6YKQ3NXFK3X5","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6YKQ3NXFK3X5ICPF","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6YKQ3NXF","created_at":"2026-05-18T12:32:11Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:6YKQ3NXFK3X5ICPFPH2G4CXGYA","target":"record","payload":{"canonical_record":{"source":{"id":"1805.11549","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-29T15:56:06Z","cross_cats_sorted":[],"title_canon_sha256":"d66387fdc1951ec6bb0bd91493cc0c07e742a512fd296fddc60a47c8b739a5f5","abstract_canon_sha256":"1411fdc084bf594cd3b488f07fb31b46edb43c2f1f430cfc0f855925e2759620"},"schema_version":"1.0"},"canonical_sha256":"f6150db6e556efd409e579f46e0ae6c025ac9caf121545efbf2c741422bc1272","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:34.039534Z","signature_b64":"XoBxN/XIdR8Qrt7iA88arRf+TH2AuAq5XK+UGRUwYCaOSPIlSEemSoC9ujqv6KRBDg/g5skM1rVNLv8DKbsyDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f6150db6e556efd409e579f46e0ae6c025ac9caf121545efbf2c741422bc1272","last_reissued_at":"2026-05-18T00:04:34.038961Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:34.038961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.11549","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":"2026-05-18T00:04:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kZ9KKpRe/BY7fsvrpo6QE9rqG92pE9Gw8pupPLJcVMtGDbIWYPPvDSGDFjHvUM/8hXmGkPzf+AO9+/tWD1kkDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:27:02.041411Z"},"content_sha256":"3cc67a3a9eb5ac40545bec1ec02731c89f6fb25ec74ed0f0150667e4b17a05cc","schema_version":"1.0","event_id":"sha256:3cc67a3a9eb5ac40545bec1ec02731c89f6fb25ec74ed0f0150667e4b17a05cc"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:6YKQ3NXFK3X5ICPFPH2G4CXGYA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Silvia Frassu","submitted_at":"2018-05-29T15:56:06Z","abstract_excerpt":"In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions as $L^{\\infty}$-bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a $C^0$-topology weighted with a suitable power of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.11549","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":"2026-05-18T00:04:34Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"M14CPeTyn8irmvGR6+VvJmA6Wtez8l1Crv/wIKgf6uFk98BOY/4aDFyIc8xaNIh6MnIOSufBGZP7b8NSPJqcBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T18:27:02.042190Z"},"content_sha256":"456e1ee4244d0ba8fd966f66175f49f69228dab66343fc2e660670a98a00b0c5","schema_version":"1.0","event_id":"sha256:456e1ee4244d0ba8fd966f66175f49f69228dab66343fc2e660670a98a00b0c5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA/bundle.json","state_url":"https://pith.science/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA/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":"2026-06-08T18:27:02Z","links":{"resolver":"https://pith.science/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA","bundle":"https://pith.science/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA/bundle.json","state":"https://pith.science/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6YKQ3NXFK3X5ICPFPH2G4CXGYA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6YKQ3NXFK3X5ICPFPH2G4CXGYA","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":"1411fdc084bf594cd3b488f07fb31b46edb43c2f1f430cfc0f855925e2759620","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-29T15:56:06Z","title_canon_sha256":"d66387fdc1951ec6bb0bd91493cc0c07e742a512fd296fddc60a47c8b739a5f5"},"schema_version":"1.0","source":{"id":"1805.11549","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.11549","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"arxiv_version","alias_value":"1805.11549v2","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.11549","created_at":"2026-05-18T00:04:34Z"},{"alias_kind":"pith_short_12","alias_value":"6YKQ3NXFK3X5","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_16","alias_value":"6YKQ3NXFK3X5ICPF","created_at":"2026-05-18T12:32:11Z"},{"alias_kind":"pith_short_8","alias_value":"6YKQ3NXF","created_at":"2026-05-18T12:32:11Z"}],"graph_snapshots":[{"event_id":"sha256:456e1ee4244d0ba8fd966f66175f49f69228dab66343fc2e660670a98a00b0c5","target":"graph","created_at":"2026-05-18T00:04:34Z","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":"In this paper we study an equation driven by a nonlocal anisotropic operator with homogeneous Dirichlet boundary conditions. We find at least three non trivial solutions: one positive, one negative and one of unknown sign, using variational methods and Morse theory. We present some results about regularity of solutions as $L^{\\infty}$-bound and Hopf's lemma, for the latter we first consider a non negative nonlinearity and then a strictly negative one. Moreover, we prove that, for the corresponding functional, local minimizers with respect to a $C^0$-topology weighted with a suitable power of t","authors_text":"Silvia Frassu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-29T15:56:06Z","title":"Nonlinear Dirichlet problem for the nonlocal anisotropic operator $L_K$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.11549","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":"sha256:3cc67a3a9eb5ac40545bec1ec02731c89f6fb25ec74ed0f0150667e4b17a05cc","target":"record","created_at":"2026-05-18T00:04:34Z","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":"1411fdc084bf594cd3b488f07fb31b46edb43c2f1f430cfc0f855925e2759620","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-29T15:56:06Z","title_canon_sha256":"d66387fdc1951ec6bb0bd91493cc0c07e742a512fd296fddc60a47c8b739a5f5"},"schema_version":"1.0","source":{"id":"1805.11549","kind":"arxiv","version":2}},"canonical_sha256":"f6150db6e556efd409e579f46e0ae6c025ac9caf121545efbf2c741422bc1272","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f6150db6e556efd409e579f46e0ae6c025ac9caf121545efbf2c741422bc1272","first_computed_at":"2026-05-18T00:04:34.038961Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:34.038961Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XoBxN/XIdR8Qrt7iA88arRf+TH2AuAq5XK+UGRUwYCaOSPIlSEemSoC9ujqv6KRBDg/g5skM1rVNLv8DKbsyDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:34.039534Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.11549","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3cc67a3a9eb5ac40545bec1ec02731c89f6fb25ec74ed0f0150667e4b17a05cc","sha256:456e1ee4244d0ba8fd966f66175f49f69228dab66343fc2e660670a98a00b0c5"],"state_sha256":"f4831261ac042a60d2ada8f98cd936584b181c8543b4a95eebfb8a1ef677e29f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"qBYwM79hXYB67IhKCTzfwqms0tf3LfGIExzqIkQcagR9VMOyXtZMELnIFCsxPjyCxMsiYNlbntQRteOZK6/uDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T18:27:02.046095Z","bundle_sha256":"1b53a33f64c32e29478068eef2af37d48a2d1b776aa0efe1b951ec9d3a1fb487"}}