{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:IUYA7NLGJOFYMTVQJPHGEVZLYJ","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":"8ef3e768db30f4cc6219405eab13cefd5e4fef639833032a85b1d0a9970b208b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-20T10:28:04Z","title_canon_sha256":"cc5a024e2092f9f22d95f5800b73ec7f8a1a48361df4e8400717ec42928be66f"},"schema_version":"1.0","source":{"id":"1507.05435","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.05435","created_at":"2026-05-18T01:36:36Z"},{"alias_kind":"arxiv_version","alias_value":"1507.05435v1","created_at":"2026-05-18T01:36:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.05435","created_at":"2026-05-18T01:36:36Z"},{"alias_kind":"pith_short_12","alias_value":"IUYA7NLGJOFY","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_16","alias_value":"IUYA7NLGJOFYMTVQ","created_at":"2026-05-18T12:29:25Z"},{"alias_kind":"pith_short_8","alias_value":"IUYA7NLG","created_at":"2026-05-18T12:29:25Z"}],"graph_snapshots":[{"event_id":"sha256:aa3fa0f0fe44849c4117959baf2d0cd1a27a5eec3b179f940cfafa2ee8c55b31","target":"graph","created_at":"2026-05-18T01:36:36Z","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":"This work is devoted to the study of the boundary value problem \\begin{eqnarray}\\nonumber (-1)^\\alpha \\Delta^\\alpha u = (-1)^k S_k[u] + \\lambda f, \\qquad x &\\in& \\Omega \\subset \\mathbb{R}^N, \\\\ \\nonumber u = \\partial_n u = \\partial_n^2 u = \\cdots = \\partial_n^{\\alpha-1} u = 0, \\qquad x &\\in& \\partial \\Omega, \\end{eqnarray} where the $k-$Hessian $S_k[u]$ is the $k^{\\mathrm{th}}$ elementary symmetric polynomial of eigenvalues of the Hessian matrix and the datum $f$ obeys suitable summability properties. We prove the existence of at least two solutions, of which at least one is isolated, strictly","authors_text":"Carlos Escudero","cross_cats":["math.FA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-20T10:28:04Z","title":"On polyharmonic regularizations of $k-$Hessian equations: Variational methods"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.05435","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":"sha256:24f0936fba1625f7d2fc4b29e20023b4ab666a39b8666ee5edab914a96622855","target":"record","created_at":"2026-05-18T01:36:36Z","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":"8ef3e768db30f4cc6219405eab13cefd5e4fef639833032a85b1d0a9970b208b","cross_cats_sorted":["math.FA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-07-20T10:28:04Z","title_canon_sha256":"cc5a024e2092f9f22d95f5800b73ec7f8a1a48361df4e8400717ec42928be66f"},"schema_version":"1.0","source":{"id":"1507.05435","kind":"arxiv","version":1}},"canonical_sha256":"45300fb5664b8b864eb04bce62572bc25b702cced2be5b8b5176ef8a2437bd8e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"45300fb5664b8b864eb04bce62572bc25b702cced2be5b8b5176ef8a2437bd8e","first_computed_at":"2026-05-18T01:36:36.810194Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:36.810194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CuPx2TefxTvC9IKc73aTOJzM9ZU8/Qmiflga+PRI6ATBiAYzoIYxmXktgs1wACkbzC6k7lU6d+hpnLYdto/vDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:36.810814Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.05435","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24f0936fba1625f7d2fc4b29e20023b4ab666a39b8666ee5edab914a96622855","sha256:aa3fa0f0fe44849c4117959baf2d0cd1a27a5eec3b179f940cfafa2ee8c55b31"],"state_sha256":"198d8db58edc955109ebf73c7676b18e8953c87de3e068209383fae1ca1956b6"}