{"paper":{"title":"Non-local functionals related to the total variation and connections with Image Processing","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Haim Brezis, Hoai-Minh Nguyen","submitted_at":"2016-08-29T19:56:53Z","abstract_excerpt":"We present new results concerning the approximation of the total variation, $\\int_{\\Omega} |\\nabla u|$, of a function $u$ by non-local, non-convex functionals of the form $$ \\Lambda_\\delta u = \\int_{\\Omega} \\int_{\\Omega} \\frac{\\delta \\varphi \\big( |u(x) - u(y)|/ \\delta\\big)}{|x - y|^{d+1}} \\, dx \\, dy, $$ as $\\delta \\to 0$, where $\\Omega$ is a domain in $\\mathrm{R}^d$ and $\\varphi: [0, + \\infty) \\to [0, + \\infty)$ is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi's concept of Gamma-con"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.08204","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"}