{"paper":{"title":"An existence theory for nonlinear equations on metric graphs via energy methods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Matthias Hofmann","submitted_at":"2019-09-17T14:45:41Z","abstract_excerpt":"The purpose of this paper is to develop a general existence theory for constrained minimization problems for functionals defined on function spaces on metric measure spaces $(\\mathcal M, d, \\mu)$. We apply this theory to functionals defined on metric graphs $\\mathcal G$, in particular $L^2$-constrained minimization problems for functionals of the form $$E(u) = \\frac{1}{2} a(u,u) - \\frac{1}{q}\\int_{\\mathcal K} |u|^q \\, \\mathrm dx,$$ where $q>2$, $a(\\cdot, \\cdot)$ is a suitable symmetric sesquilinear form on some function space on $\\mathcal G$ and $\\mathcal K \\subseteq \\mathcal G$ is given.\n  We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.07856","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1909.07856/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}