{"paper":{"title":"On the $\\Gamma$-limit of singular perturbation problems with optimal profiles which are not one-dimensional. Part I: The upper bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Arkady Poliakovsky","submitted_at":"2011-12-10T21:55:14Z","abstract_excerpt":"In Part I we construct the upper bound, in the spirit of $\\Gamma$- $\\limsup$, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form $$E_\\e(v):=\\int_\\Omega \\frac{1}{\\e}F\\Big(\\e^n\\nabla^n v,...,\\e\\nabla v,v\\Big)dx\\quad\\text{for} v:\\Omega\\subset\\R^N\\to\\R^k \\text{such that} A\\cdot\\nabla v=0,$$ where the function $F\\geq 0$ and $A:\\R^{k\\times N}\\to\\R^m$ is a prescribed linear operator (for example, $A:\\equiv 0$, $A\\cdot\\nabla v:=\\text{curl}v$ and $A\\cdot\\nabla v=\\text{div}\\,v$) which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.2305","kind":"arxiv","version":12},"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"}