{"paper":{"title":"Lions' Maximal Regularity Problem for Divergence-Form Differential Operators: Failure at the $\\frac{1}{2}$-H\\\"older Endpoint","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.FA"],"primary_cat":"math.AP","authors_text":"Lukas Niebel","submitted_at":"2026-08-11T17:52:02Z","abstract_excerpt":"In this work we give a counterexample to maximal $\\mathrm{L}^2$-regularity in Lions' problem for divergence-form differential operators. On a bounded interval, we construct a bounded, uniformly elliptic, real scalar diffusion coefficient that is $\\frac{1}{2}$-H\\\"older continuous in time with values in spatial $\\mathrm{L}^\\infty$. It can be chosen arbitrarily close to the constant coefficient of the heat equation. For zero initial data and a forcing term that is continuous in time with square-integrable spatial values, the unique Lions variational solution has a time derivative that is not squa"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.11194","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/2608.11194/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"}