{"paper":{"title":"The regularity of the positive part of functions in $L^2(I; H^1(\\Omega)) \\cap H^1(I; H^1(\\Omega)^*)$ with applications to parabolic equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Daniel Wachsmuth","submitted_at":"2016-04-15T08:01:04Z","abstract_excerpt":"Let $u\\in L^2(I; H^1(\\Omega))$ with $\\partial_t u\\in L^2(I; H^1(\\Omega)^*)$ be given. Then we show by means of a counter-example that the positive part $u^+$ of $u$ has less regularity, in particular it holds $\\partial_t u^+ \\not\\in L^1(I; H^1(\\Omega)^*)$ in general. Nevertheless, $u^+$ satisfies an integration-by-parts formula, which can be used to prove non-negativity of weak solutions of parabolic equations."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.04392","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":""},"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"}