{"paper":{"title":"Optimal Asymptotic Behavior at Infinity for Solutions of the Supercritical Lagrangian Mean Curvature Equation in Exterior Domains","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"Solutions to the supercritical Lagrangian mean curvature equation in two dimensions converge to quadratic polynomials at infinity under merely Lipschitz perturbations that decay at any positive rate.","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jiguang Bao, Qinfeng Jiang","submitted_at":"2026-04-29T03:04:02Z","abstract_excerpt":"We study the asymptotic behavior at infinity of solutions to the supercritical Lagrangian mean curvature equation \\[ \\sum_{i=1}^n \\arctan \\lambda_i(D^2u)=\\theta+f(x) \\] on exterior domains in \\(\\mathbb R^n\\), \\(n\\ge 2\\), where \\(|\\theta|>((n-2)\\pi)/2\\). The perturbation \\(f\\) is assumed to be locally Lipschitz near infinity and to satisfy a decay condition with rate \\(\\beta>0\\). The main new ingredient is a scale-dependent difference quotient argument, combined with a nonlocal potential method, which avoids differentiating \\(f\\) twice and yields quantitative Hessian convergence under only Lips"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"This work generalizes the convergence results in [BJ2026], where f is required to be at least C^3 and β>2. Moreover, all asymptotic results established in this paper are optimal.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The perturbation f is Lipschitz continuous and satisfies f(x) = O(|x|^{-β}) for some β > 0 at infinity, with |θ| in (0, π) a constant phase, and the equation holds on exterior domains in R^2.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Solutions to the supercritical Lagrangian mean curvature equation in 2D exterior domains exhibit optimal asymptotic behavior at infinity under Lipschitz perturbations decaying at any positive rate.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Solutions to the supercritical Lagrangian mean curvature equation in two dimensions converge to quadratic polynomials at infinity under merely Lipschitz perturbations that decay at any positive rate.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"325bd6c0bfd5463333efcc15edef454bd2da0fd38b9f7428ca58dab5c8c7d4a0"},"source":{"id":"2604.26246","kind":"arxiv","version":2},"verdict":{"id":"ef7c65a0-6f44-48d0-96cb-3728aef8af3e","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-07T13:31:39.627365Z","strongest_claim":"This work generalizes the convergence results in [BJ2026], where f is required to be at least C^3 and β>2. Moreover, all asymptotic results established in this paper are optimal.","one_line_summary":"Solutions to the supercritical Lagrangian mean curvature equation in 2D exterior domains exhibit optimal asymptotic behavior at infinity under Lipschitz perturbations decaying at any positive rate.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The perturbation f is Lipschitz continuous and satisfies f(x) = O(|x|^{-β}) for some β > 0 at infinity, with |θ| in (0, π) a constant phase, and the equation holds on exterior domains in R^2.","pith_extraction_headline":"Solutions to the supercritical Lagrangian mean curvature equation in two dimensions converge to quadratic polynomials at infinity under merely Lipschitz perturbations that decay at any positive rate."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.26246/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"ai_meta_artifact","ran_at":"2026-05-21T00:39:53.730893Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T20:22:55.601406Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"c6062efecce5dd05fe69f08ff67b41630463ee5e05919aced44468fbc00716b4"},"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"}