{"paper":{"title":"Cascade of minimizers for a nonlocal isoperimetric problem in thin domains","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Massimiliano Morini, Peter Sternberg","submitted_at":"2013-09-03T07:13:40Z","abstract_excerpt":"For $\\Omega_\\e=(0,\\e)\\times (0,1)$ a thin rectangle, we consider minimization of the two-dimensional nonlocal isoperimetric problem given by \\[ \\inf_u E^{\\gamma}_{\\Omega_\\e}(u)\\] where \\[ E^{\\gamma}_{\\Omega_\\e}(u):= P_{\\Omega_\\e}(\\{u(x)=1\\})+\\gamma\\int_{\\Omega_\\e}\\abs{\\nabla{v}}^2\\,dx \\] and the minimization is taken over competitors $u\\in BV(\\Omega_\\e;\\{\\pm 1\\})$ satisfying a mass constraint $\\fint_{\\Omega_\\e}u=m$ for some $m\\in (-1,1)$. Here $P_{\\Omega_\\e}(\\{u(x)=1\\})$ denotes the perimeter of the set $\\{u(x)=1\\}$ in $\\Omega_\\e$, $\\fint$ denotes the integral average and $v$ denotes the solut"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.0597","kind":"arxiv","version":2},"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"}