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In the case when $f\\geq 0$ in $D$ the inequality is optimal for any domain and for any values of $\\|f\\|_1$ and $\\|f\\|_\\infty.$ We also show that $$ \\sigma_D(t)\\leq\\sigma_B("},"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":"1707.07557","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2017-07-24T13:44:48Z","cross_cats_sorted":[],"title_canon_sha256":"62ee53a45e454d799df976ea5ec16625ceea32d87b7f954f30d42679df757326","abstract_canon_sha256":"c09446e1f28d72e70286a1cdc5115bc4420ee28261603ff91609dd2da4189cb6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:41.863365Z","signature_b64":"3v3ybFclE6T5W5pVMJioJNxBGbD5OUANuYlLAqHsHzTCFpzSPNJYVwvYLga2J9UG/M/PQuKp3mvdF+1TETjSDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"107bedd9e2a5da5273c7504deb164967b60d9f259349c6f3b16d59ce15a99e2a","last_reissued_at":"2026-05-18T00:39:41.862798Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:41.862798Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the $L^\\infty-$maximization of the solution of Poisson's equation: Brezis-Gallouet-Wainger type inequalities and applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Davit Harutyunyan, Hayk Mikayelyan","submitted_at":"2017-07-24T13:44:48Z","abstract_excerpt":"For the solution of the Poisson problem with an $L^\\infty$ right hand side \\begin{equation*} \\begin{cases} -\\Delta u(x) = f (x) & \\mbox{in } D,\n  u=0 & \\mbox{on } \\partial D, \\end{cases} \\end{equation*} we derive an optimal estimate of the form $$ \\|u\\|_\\infty\\leq \\|f\\|_\\infty \\sigma_D(\\|f\\|_1/\\|f\\|_\\infty), $$ where $\\sigma_D$ is a modulus of continuity defined in the interval $[0, |D|]$ and depends only on the domain $D$. 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