{"paper":{"title":"Asymptotic behavior of the $W^{1/q,q}$-norm of mollified $BV$ functions and applications to singular perturbation problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.FA","math.MP"],"primary_cat":"math.AP","authors_text":"Arkady Poliakovsky","submitted_at":"2018-12-15T21:18:20Z","abstract_excerpt":"Motivated by results of Figalli and Jerison and Hern\\'andez, we prove the following formula: \\begin{equation*} \\lim_{\\epsilon\\to 0^+}\\frac{1}{|\\ln{\\epsilon}|}\\big\\|\\eta_\\epsilon*u\\big\\|^q_{W^{1/q,q}(\\Omega)}= C_0\\int_{J_u}\\Big|u^+(x)-u^-(x)\\Big|^qd\\mathcal{H}^{N-1}(x), \\end{equation*} where $\\Omega\\subset\\mathbb{R}^N$ is a regular domain, $u\\in BV(\\Omega)\\cap L^\\infty$, $q>1$ and $\\eta_\\epsilon(z)=\\epsilon^{-N}\\eta(z/\\epsilon)$ is a smooth mollifier. In addition, we apply the above formula to the study of certain singular perturbation problems."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1812.06358","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"}