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At the difference with the case $p = 2,$ which is simpler, we call here the weak reverse H\\\"older inequality. This estimates show that the solution of Robin problem converges strongly to the solution of Dirichlet (res"},"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":"1805.09519","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-05-24T06:19:50Z","cross_cats_sorted":[],"title_canon_sha256":"4f85811eeee6db1ed91a6b9c287233319e84c0b8dd01a2b5954600cd54e5bda1","abstract_canon_sha256":"60ceed43a918ecba4314f0c30425266f655735276b8c29c156219a2bb92570f6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:05:09.213273Z","signature_b64":"MTFqBFhh65RTL+r5yLz9Dz/AD7ZV9ucrqrMkE43ij2x6FzS2VW+I8IcCLfQ+0ysWfJtmeGWN6O8BrVDgr8PlDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2be1b00cac584847d8b35e66cca3cec57dceed272eb766013cab66f20a3b3f12","last_reissued_at":"2026-05-18T00:05:09.212844Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:05:09.212844Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Uniform $W^{1,p}$ estimate for elliptic operator with Robin boundary condition in $\\mathcal{C}^1$ domain","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Amrita Ghosh, Carlos Conca, Cherif Amrouche, Tuhin Ghosh","submitted_at":"2018-05-24T06:19:50Z","abstract_excerpt":"We consider the Robin boundary value problem $\\mathrm{div} (A \\nabla u) = \\mathrm{div} \\mathbf{f}+F$ in $\\Omega$, $\\mathcal{C}^1$ domain, with $(A \\nabla u - \\mathbf{f})\\cdot \\mathbf{n} + \\alpha u = g$ on $\\Gamma$, where the matrix $A$ belongs to $VMO (\\mathbb{R}^3) $, and discover the uniform estimates on $\\|u\\|_{W^{1,p}(\\Omega)}$, with $1 < p < \\infty$, independent on $\\alpha$. At the difference with the case $p = 2,$ which is simpler, we call here the weak reverse H\\\"older inequality. 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