{"paper":{"title":"A geometric approach to regularity for nonlinear free boundary problems with finite Morse index","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram L. Karakhanyan","submitted_at":"2017-02-01T21:49:09Z","abstract_excerpt":"Let $u$ be a weak solution of the free boundary problem $$\\mathcal L u=\\lambda_0 \\mathcal H^1\\lfloor\\partial\\{u>0\\}, u\\ge 0,$$ where $\\mathcal L u={\\text{div}}(g(\\nabla u)\\nabla u)$ is a quasilinear elliptic operator and $g(\\xi)$ is a given function of $\\xi$ satisfying some structural conditions. We prove that the free boundary $\\partial\\{ u>0\\}$ is continuously differentiable in $\\mathbb R^2$, provided that $\\partial\\{ u>0\\}$ has locally finite connectivity. Moreover, we show that the free boundaries of weak solutions with finite $\\it{Morse \\ index}$ must have finite connectivity. The weak so"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.00465","kind":"arxiv","version":3},"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"}