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We prove here pointwise estimates on $G_n(x,y)$, $\\nabla_x G_n(x,y)$, $\\nabla_y G_n(x,y)$ and $\\nabla_x \\nabla_y G_n(x,y)$ in dimensions $d \\geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. These results also apply for systems."},"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":"1807.09062","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2018-07-24T12:23:05Z","cross_cats_sorted":[],"title_canon_sha256":"9c29858eb1cd55266611265f8a1e64890503e6e9f631f8e671fb7db229b1ea25","abstract_canon_sha256":"09c2f77c6d93e7c245102fb6d9156b63c329e08617615698a11fd2a3ead0295c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:09:56.899377Z","signature_b64":"i4NEoAQgofHvIReM0QPY96X3jGhe6cEFxPs9ttBfMieSC8pYv/CN1WiSaH4J7st690ZjtGRmVmWCGgKOXzhmDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3f1ff16d7fdb0edb5880d7878dd2f7814bbcfc61b83d55972b103dcc35e0a8ba","last_reissued_at":"2026-05-18T00:09:56.898753Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:09:56.898753Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Decomposition and pointwise estimates of periodic Green functions of some elliptic equations with periodic oscillatory coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Marc Josien","submitted_at":"2018-07-24T12:23:05Z","abstract_excerpt":"This article is about the $\\mathbb{Z}^d$-periodic Green function $G_n(x,y)$ of the multiscale elliptic operator $Lu=-{\\rm div}\\left( A(n\\cdot) \\cdot \\nabla u \\right)$, where $A(x)$ is a $\\mathbb{Z}^d$-periodic, coercive, and H\\\"older continuous matrix, and $n$ is a large integer. We prove here pointwise estimates on $G_n(x,y)$, $\\nabla_x G_n(x,y)$, $\\nabla_y G_n(x,y)$ and $\\nabla_x \\nabla_y G_n(x,y)$ in dimensions $d \\geq 2$. Moreover, we derive an explicit decomposition of this Green function, which is of independent interest. 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