{"paper":{"title":"Domains in metric measure spaces with boundary of positive mean curvature, and the Dirichlet problem for functions of least gradient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Gareth Speight, Lukas Maly, Nageswari Shanmugalingam, Panu Lahti","submitted_at":"2017-06-22T22:16:15Z","abstract_excerpt":"We study the geometry of domains in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\\'e inequality. We propose a notion of \\emph{domain with boundary of positive mean curvature} and prove that, for such domains, there is always a solution to the Dirichlet problem for least gradients with continuous boundary data. Here \\emph{least gradient} is defined as minimizing total variation (in the sense of BV functions) and boundary conditions are satisfied in the sense that the \\emph{boundary trace} of the solution exists and agrees with the given boundary data. "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1706.07505","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"}