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The above system is said to be symmetric if the matrix of gradient of all components of $H$ is symmetric. It seems that this concept is crucial to prove Liouville theorems, when $\\Omega=\\mathbb R^n$, and regularity results, when $\\Omega=B_1$, for stable solutions of the above system for a general nonlinearity $H \\in C^1(\\math"},"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":"1410.1831","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-10-07T18:13:55Z","cross_cats_sorted":[],"title_canon_sha256":"d26f9acff6ba3f03d9924ed2982b6522b008ccb8e9085aa4202a3ecaceb798f8","abstract_canon_sha256":"247560c8517dca44a4bee15b79d165cb1c310807aa5620c8faf9badf4bfa5186"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:40:54.651027Z","signature_b64":"0na1FZRCwdGb35rdm5nTzRVTTpAzPSzJVuEf43xYZ/31AaZwSoSrzDBE7Z+Vk4BdOZdtv9x9M6TC8TXKV6JHAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da8ab870e78a8b7926391011167e2051137e378924dd871067f658ebee104a86","last_reissued_at":"2026-05-18T02:40:54.650509Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:40:54.650509Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rigidity results for stable solutions of symmetric systems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Mostafa Fazly","submitted_at":"2014-10-07T18:13:55Z","abstract_excerpt":"We study stable solutions of the following nonlinear system $$ -\\Delta u = H(u) \\quad \\text{in} \\ \\ \\Omega$$ where $u:\\mathbb R^n\\to \\mathbb R^m$, $H:\\mathbb R^m\\to \\mathbb R^m$ and $\\Omega$ is a domain in $\\mathbb R^n$. 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