{"paper":{"title":"Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals","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":"2017-03-13T00:37:40Z","abstract_excerpt":"Motivated by the formula, due to Bourgain, Brezis and Mironescu, \\begin{equation*}\n  \\lim_{\\varepsilon\\to 0^+} \\int_\\Omega\\int_\\Omega \\frac{|u(x)-u(y)|^q}{|x-y|^q}\\,\\rho_\\varepsilon(x-y)\\,dx\\,dy=K_{q,N}\\|\\nabla u\\|_{L^{q}}^q\\,, \\end{equation*} that characterizes the functions in $L^q$ that belong to $W^{1,q}$ (for $q>1$) and $BV$ (for $q=1$), respectively, we study what happens when one replaces the denominator in the expression above by $|x-y|$. It turns out that, for $q>1$ the corresponding functionals \"see\" only the jumps of the $BV$ function. We further identify the function space relevant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.04208","kind":"arxiv","version":3},"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"}