{"paper":{"title":"Relaxation and optimization for linear-growth convex integral functionals under PDE constraints","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Adolfo Arroyo-Rabasa","submitted_at":"2016-03-03T22:57:56Z","abstract_excerpt":"We give necessary and sufficient conditions for minimality of generalized minimizers for linear-growth functionals of the form \\[ \\mathcal F[u] := \\int_\\Omega f(x,u(x)) \\, \\text{d}x, \\qquad u:\\Omega \\subset \\mathbb R^N\\to \\mathbb R^d, \\] where $u$ is an integrable function satisfying a general PDE constraint. Our analysis is based on two ideas: a relaxation argument into a subspace of the space of bounded vector-valued Radon measures $\\mathcal M(\\Omega;\\mathbb R^d)$, and the introduction of a set-valued pairing in $\\mathcal M(\\Omega;\\mathbb R^N) \\times {\\rm L}^\\infty(\\Omega;\\mathbb R^N)$. By t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01310","kind":"arxiv","version":3},"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"}