{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:7S34FR3VJD52SO7GZINH57Z3VN","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":"2333bed26f43399aa36c6abd9029fe4feca05a65d509344a98edd87ab563b142","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-27T08:05:41Z","title_canon_sha256":"d9c237cd2bde04f11aeeef2737ca74898c43463a9dea80ffbbbdd61f09c2a262"},"schema_version":"1.0","source":{"id":"1207.6480","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1207.6480","created_at":"2026-05-18T00:17:36Z"},{"alias_kind":"arxiv_version","alias_value":"1207.6480v2","created_at":"2026-05-18T00:17:36Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1207.6480","created_at":"2026-05-18T00:17:36Z"},{"alias_kind":"pith_short_12","alias_value":"7S34FR3VJD52","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"7S34FR3VJD52SO7G","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"7S34FR3V","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:4331579f916e93e669a439b081fc9cc0d0d4e6ac7bd18c3ad85a2e6019d3b5c9","target":"graph","created_at":"2026-05-18T00:17:36Z","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":"For every $f \\in L^N(\\Omega)$ defined in an open bounded subset $\\Omega$ of $\\mathbb{R}^N$, we prove that a solution $u \\in W_0^{1, 1}(\\Omega)$ of the $1$-Laplacian equation ${-}\\mathrm{div}{(\\frac{\\nabla u}{|\\nabla u|})} = f$ in $\\Omega$ satisfies $\\nabla u = 0$ on a set of positive Lebesgue measure. The same property holds if $f \\not\\in L^N(\\Omega)$ has small norm in the Marcinkiewicz space of weak-$L^{N}$ functions or if $u$ is a BV minimizer of the associated energy functional. The proofs rely on Stampacchia's truncation method.","authors_text":"Augusto C. Ponce, Luigi Orsina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-27T08:05:41Z","title":"Flat solutions of the 1-Laplacian equation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1207.6480","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":"sha256:6adda7499e4541d193b825856738cdcab19fdc7417e44d422270ac18b2d24f9f","target":"record","created_at":"2026-05-18T00:17:36Z","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":"2333bed26f43399aa36c6abd9029fe4feca05a65d509344a98edd87ab563b142","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2012-07-27T08:05:41Z","title_canon_sha256":"d9c237cd2bde04f11aeeef2737ca74898c43463a9dea80ffbbbdd61f09c2a262"},"schema_version":"1.0","source":{"id":"1207.6480","kind":"arxiv","version":2}},"canonical_sha256":"fcb7c2c77548fba93be6ca1a7eff3bab589db643c1916c724febaf877054977a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fcb7c2c77548fba93be6ca1a7eff3bab589db643c1916c724febaf877054977a","first_computed_at":"2026-05-18T00:17:36.051090Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:36.051090Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"flGZCTIkkv0EPdiOfV90Nl7QXGGK2F5S8vUSPqBoHYHHadkgwKYRvrln6eNH6noc3jS/u4NpfVNfG+IfBAubCw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:36.051691Z","signed_message":"canonical_sha256_bytes"},"source_id":"1207.6480","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6adda7499e4541d193b825856738cdcab19fdc7417e44d422270ac18b2d24f9f","sha256:4331579f916e93e669a439b081fc9cc0d0d4e6ac7bd18c3ad85a2e6019d3b5c9"],"state_sha256":"a7b0671e41c1b5f74ff2b70cb8954a0f4db726313b749b0524bcbecff4c9a0e1"}