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We will prove the existence of constants $\\lambda_N\\ge\\lambda_N^{\\ast}\\ge\\lambda^{\\ast}(1+\\chi\\int_{\\Omega}\\frac{dx}{1-w_{\\ast}})^2$ for the nonlocal MEMS equation $-\\Delta v=\\lam/(1-v)^2(1+\\chi\\int_{\\Omega}1/(1-v)dx)^2$ in $\\Omega$, $v=0$ on $\\1\\Omega$, such that a solution exists for any $0\\le\\lambda<\\lambda_N^{\\ast}$ and no solution exists for any $\\lambda>\\lambda_N$ where $\\lambda^{\\ast}$ is the pull-in voltage and $w_{\\ast}$ is the limit of the minimal solution of $-\\Delta v=\\lam/(1-v)^2$ in $\\Omega$ wit"},"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":"0809.4209","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2008-09-24T15:05:30Z","cross_cats_sorted":[],"title_canon_sha256":"516bec2d7641658121c11cf112b60074067fefd6458a108eeb77d9aaaab88f1b","abstract_canon_sha256":"fc2fb6d45a5e165103f73053f7af9c37c44744784e3fd6bc6cb65045ce9a1f6f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:42:12.561855Z","signature_b64":"lst0RVNZO2ewv82MUm5ue7dYD2eOJnAl0Sskk6piVd4fe/hJfEkLMc0JB9ODE9bfdEWdRbX7eiQEhnkLzcQuBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d697dce89419dde53175b444b4e70f748ecbc832ba3d1172343173ed1f859ab2","last_reissued_at":"2026-05-18T04:42:12.561268Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:42:12.561268Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Existence and dynamic properties of a parabolic nonlocal MEMS equation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Kin Ming Hui","submitted_at":"2008-09-24T15:05:30Z","abstract_excerpt":"Let $\\Omega\\subset\\mathbb{R}^n$ be a $C^2$ bounded domain and $\\chi>0$ be a constant. 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