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We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\\partial\\{u>0\\}$ can be seen as a rectifiable $n-1$ varifold. If the free boundary is a Lipschitz multigraph then we show that the first variation of this varifold is bounded and use Allard's monotonicity formula to"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1811.07620","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-11-19T11:23:14Z","cross_cats_sorted":[],"title_canon_sha256":"0cd176303581d1b9063f8e4991522b50373ee3ebc65913d59e7c7dcf2090a85e","abstract_canon_sha256":"78883dd751168bd9af6c86d9043853ae151f24389f78b1c2680d3a54bca56faf"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:59:31.983523Z","signature_b64":"KiBFefRaYPK70EaB0kIxJmaNI81TFwdZdl77QG7aTRX3K4A9Mi97wpJtWkZEE0UFiRLTXnCYi3bxW02DfmNUDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bb55262e300e043ef6a296d9b99aa6f43e0e43b22f3b88bd8e2d05a63e0ceefc","last_reissued_at":"2026-05-17T23:59:31.982836Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:59:31.982836Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Full and partial regularity for a class of nonlinear free boundary problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Aram Karakhanyan","submitted_at":"2018-11-19T11:23:14Z","abstract_excerpt":"In this paper we classify the nonnegative global minimizers of the functional \\[ J_F(u)=\\int_\\Omega F(|\\nabla u|^2)+\\lambda^2\\chi_{\\{u>0\\}}, \\] where $F$ satisfies some structural conditions and $\\chi_D$ is the characteristic function of a set $D\\subset \\mathbb R^n$. We compute the second variation of the energy and study the properties of the stability operator. The free boundary $\\partial\\{u>0\\}$ can be seen as a rectifiable $n-1$ varifold. 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