{"paper":{"title":"Optimal Regularity for The Signorini Problem and its Free Boundary","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"John Andersson","submitted_at":"2013-10-09T14:57:52Z","abstract_excerpt":"We will show optimal regularity for minimizers of the Signorini problem for the Lame system. In particular if $\\u=(u^1,u^2,u^3)\\in W^{1,2}(B_1^+:\\R^3)$ minimizes $$ J(\\u)=\\int_{B_1^+}|\\nabla \\u+\\nabla^\\bot \\u|^2+\\lambda\\div(\\u)^2 $$ in the convex set $$ K=\\big\\{\\u=(u^1,u^2,u^3)\\in W^{1,2}(B_1^+:\\R^3);\\; u^3\\ge 0 \\textrm{on}\\Pi, $$ $$ \\u=f\\in C^\\infty(\\partial B_1) \\textrm{on}(\\partial B_1)^+ \\big\\}, $$ where $\\lambda\\ge 0$ say.\n  Then $\\u\\in C^{1,1/2}(B_{1/2}^+)$. Moreover the free boundary, given by $$ \\Gamma_\\u=\\partial \\{x;\\;u^3(x)=0,\\; x_3=0\\}\\cap B_{1}, $$ will be a $C^{1,\\alpha}$ graph c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1310.2511","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"}