{"paper":{"title":"On shape optimization problems involving the fractional laplacian","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Anne-Laure Dalibard (DMA), David G\\'erard-Varet (IMJ)","submitted_at":"2012-02-22T14:41:25Z","abstract_excerpt":"Our concern is the computation of optimal shapes in problems involving $\\(-\\Delta)^{1/2}$. We focus on the energy $J(\\Omega)$ associated to the solution $u\\_\\Omega$ of the basic Dirichlet problem $(-\\Delta)^{1/2} u\\_\\Omega = 1$ in $\\Omega$, $ u = 0$ in $\\Omega^c$. We show that regular minimizers $\\Omega$ of this energy under a volume constraint are disks. Our proof goes through the explicit computation of the shape derivative (that seems to be completely new in the fractional context), and a refined adaptation of the moving plane method."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4920","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"}