{"paper":{"title":"On Variational Approximations For Wave Maps","license":"http://creativecommons.org/licenses/by/4.0/","headline":"Global weak solutions for wave maps into spheres exist as singular limits of minimizers to time-weighted elliptic functionals.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Changyou Wang, Zhiyuan Geng","submitted_at":"2026-05-17T16:41:41Z","abstract_excerpt":"n this paper, we revisit the existence of global weak solutions of wave maps from $\\R^n$ into the sphere $\\mathbb{S}^{L-1}$, $\\Box u\\perp T_u \\mathbb{S}^{L-1}$, by establishing it as a singular limit of maps from $\\R^n\\times \\R_+$ to $\\mathbb S^{L-1}$ that minimize elliptic regularized variational functionals that contain an exponential weight in the time direction with a small parameter $\\varepsilon$, where the initial data of the Cauchy problem serve as the boundary condition. The idea went back to De Giorgi \\cite{Giorgi1996}, which has been implemented by Serra and Tilli \\cite{Serra-Tilli20"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"The existence of global weak solutions of wave maps from R^n into S^{L-1} satisfying Box u perpendicular to T_u S^{L-1} is established as a singular limit of maps minimizing elliptic regularized variational functionals containing an exponential weight in the time direction with small parameter ε.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The paper assumes that the minimizers of the ε-regularized functionals exist for each ε > 0 and that a suitable compactness or convergence argument as ε → 0 produces a limit satisfying the wave map equation and the initial data in the weak sense (abstract, paragraph on the singular limit construction).","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Global weak solutions for wave maps from R^n to S^{L-1} are recovered as singular limits of minimizers of exponentially weighted elliptic regularized functionals with initial data as boundary conditions.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Global weak solutions for wave maps into spheres exist as singular limits of minimizers to time-weighted elliptic functionals.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"a1a07e7aeca00d723605ffeb605c264030ee45d96d87ee0a9921fd097fe7564d"},"source":{"id":"2605.17536","kind":"arxiv","version":1},"verdict":{"id":"c3bc7276-5082-46cf-b0a4-3a5b14753ef7","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T22:09:15.006385Z","strongest_claim":"The existence of global weak solutions of wave maps from R^n into S^{L-1} satisfying Box u perpendicular to T_u S^{L-1} is established as a singular limit of maps minimizing elliptic regularized variational functionals containing an exponential weight in the time direction with small parameter ε.","one_line_summary":"Global weak solutions for wave maps from R^n to S^{L-1} are recovered as singular limits of minimizers of exponentially weighted elliptic regularized functionals with initial data as boundary conditions.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The paper assumes that the minimizers of the ε-regularized functionals exist for each ε > 0 and that a suitable compactness or convergence argument as ε → 0 produces a limit satisfying the wave map equation and the initial data in the weak sense (abstract, paragraph on the singular limit construction).","pith_extraction_headline":"Global weak solutions for wave maps into spheres exist as singular limits of minimizers to time-weighted elliptic functionals."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.17536/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"doi_title_agreement","ran_at":"2026-05-19T22:31:19.598181Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T22:21:44.817991Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T21:33:23.616078Z","status":"skipped","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T21:21:57.553131Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c0f9cd22a4a53a7997978c729d5600a173f77de198db4b42a3274c1b1a61c903"},"references":{"count":10,"sample":[{"doi":"","year":1996,"title":"De Giorgi,Conjectures concerning some evolution problems","work_id":"b4e65bef-e1df-4003-8440-6036ba65cead","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1996,"title":"Freire,Global weak solutions of the wave map system to compact homogeneous spaces.Manuscripta Math","work_id":"d2f65f44-3801-4a34-8e2a-853c4f407447","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1997,"title":"A. 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