{"paper":{"title":"A Self-dual Polar Factorization for Vector Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Abbas Moameni, Nassif Ghoussoub","submitted_at":"2011-01-26T00:36:19Z","abstract_excerpt":"We show that any non-degenerate vector field $u$ in $ L^{\\infty}(\\Omega, \\R^N)$, where $\\Omega$ is a bounded domain in $\\R^N$, can be written as {equation} \\hbox{$u(x)= \\nabla_1 H(S(x), x)$ for a.e. $x \\in \\Omega$}, {equation} where $S$ is a measure preserving point transformation on $\\Omega$ such that $S^2=I$ a.e (an involution), and $H: \\R^N \\times \\R^N \\to \\R$ is a globally Lipschitz anti-symmetric convex-concave Hamiltonian. Moreover, $u$ is a monotone map if and only if $S$ can be taken to be the identity, which suggests that our result is a self-dual version of Brenier's polar decomposit"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1101.4979","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"}