{"paper":{"title":"Diffusion Coefficients Estimation for Elliptic Partial Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Albert Cohen, Andrea Bonito, Gerrit Welper, Guergana Petrova, Ronald DeVore","submitted_at":"2016-09-16T20:43:23Z","abstract_excerpt":"This paper considers the Dirichlet problem $$ -\\mathrm{div}(a\\nabla u_a)=f \\quad \\hbox{on}\\,\\,\\ D, \\qquad u_a=0\\quad \\hbox{on}\\,\\,\\partial D, $$ for a Lipschitz domain $D\\subset \\mathbb R^d$, where $a$ is a scalar diffusion function. For a fixed $f$, we discuss under which conditions is $a$ uniquely determined and when can $a$ be stably recovered from the knowledge of $u_a$.\n  A first result is that whenever $a\\in H^1(D)$, with $0<\\lambda \\le a\\le \\Lambda$ on $D$, and $f\\in L_\\infty(D)$ is strictly positive, then $$ \\|a-b\\|_{L_2(D)}\\le C\\|u_a-u_b\\|_{H_0^1(D)}^{1/6}. $$ More generally, it is sh"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05231","kind":"arxiv","version":2},"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"}