{"paper":{"title":"On stable solutions for boundary reactions: a De Giorgi-type result in dimension 4+1","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Alessio Figalli, Joaquim Serra","submitted_at":"2017-05-08T08:51:27Z","abstract_excerpt":"We prove that every bounded stable solution of \\[ (-\\Delta)^{1/2} u + f(u) =0 \\qquad \\mbox{in }\\mathbb R^3\\] is a 1D profile, i.e., $u(x)= \\phi(e\\cdot x)$ for some $e\\in \\mathbb S^2$, where $\\phi:\\mathbb R\\to \\mathbb R$ is a nondecreasing bounded stable solution in dimension one. This proves the De Giorgi conjecture in dimension $4$ for the half-Laplacian. Equivalently, we give a positive answer to the De Giorgi conjecture for boundary reactions in $\\mathbb R^{d+1}_+=\\mathbb R^{d+1}\\cap \\{x_{d+1}\\geq 0\\}$ when $d = 4$, by proving that all critical points of $$ \\int_{\\{x_{d+1\\geq 0}\\}} \\frac12 "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02781","kind":"arxiv","version":1},"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"}