{"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 $D$. 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("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.07557","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}