{"paper":{"title":"Regular Curves, Singular Graphs: Cantor Parts and the Relaxed Willmore Energy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Boris Gulyak, Hans-Christoph Grunau","submitted_at":"2026-07-14T15:36:07Z","abstract_excerpt":"One might expect that finite relaxed elastic energy rules out diffuse singularities in the derivative, leaving only absolutely continuous and jump parts. This is suggested by the role of $SBV$ in free-discontinuity problems and by interpreting jumps as vertical segments of limiting graphs. We show that it fails for the relaxed one-dimensional Willmore energy. We construct a continuous function $u\\in BV((0,1))$ with $D^c u\\neq0$ and $\\overline{\\mathcal{W}}(u)<\\infty $, so finite relaxed Willmore energy does not imply $u\\in SBV((0,1))$. The idea is to concentrate the Cantor part exactly where th"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.12899","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.12899/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}