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The operator $L$ is given by $$L(u_i (x)):= \\int_{\\mathbb R^2} [u_i(x) - u_i(z)] K(z-x) dz,$$ for some kernel $K$. The idea is to apply a linear Liouville theorem for the quotient of partial derivatives, just like in the proof of the classical De Giorgi's conjecture in lower d"},"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":"1506.03368","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2015-06-10T15:49:52Z","cross_cats_sorted":[],"title_canon_sha256":"b98c626af7e24943c024e7de9b22f6a05adba3bd85c1e6ddcd1d5904a3770c2f","abstract_canon_sha256":"c16102d3c91419092f13d4f166ca49ff14db0965e682578229ddb436ea8d524a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:55:24.910618Z","signature_b64":"SnZP20NYcNCsLn5MHLwkeYWkECqT+wQ+ot4lpMjU1BT1f8gx21FIWdac8u1SUKrWhD9VODBrZ1rEq/+VXtUcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"416886ec60833c72881765c2e4a7af48a470dbaa059fcc9031ae11472a1c964e","last_reissued_at":"2026-05-18T01:55:24.909867Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:55:24.909867Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"One-dimensional symmetry for integral systems in two dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2015-06-10T15:49:52Z","abstract_excerpt":"The purpose of this brief paper is to prove De Giorgi type results for stable solutions of the following nonlocal system of integral equations in two dimensions $$ L(u_i) = H_i(u) \\quad \\text{in} \\ \\ \\mathbb R^2 , $$ where $u=(u_i)_{i=1}^m$ for $u_i: \\mathbb R^n\\to \\mathbb R$, $H=(H_i)_{i=1}^m$ is a general nonlinearity. 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