{"paper":{"title":"A Comparison Principle for a Sobolev Gradient Semi-Flow","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Enrico Valdinoci, Rafael de la Llave, Timothy Blass","submitted_at":"2009-10-12T19:14:54Z","abstract_excerpt":"We consider gradient descent equations for energy functionals of the type S(u) = 1/2 < u(x), A(x)u(x) >_{L^2} + \\int_{\\Omega} V(x,u) dx, where A is a uniformly elliptic operator of order 2, with smooth coefficients. The gradient descent equation for such a functional depends on the metric under consideration.\n  We consider the steepest descent equation for S where the gradient is an element of the Sobolev space H^{\\beta}, \\beta \\in (0,1), with a metric that depends on A and a positive number \\gamma > \\sup |V_{22}|. We prove a weak comparison principle for such a gradient flow.\n  We extend our "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0910.2214","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"}