{"paper":{"title":"Applications of Fourier analysis in homogenization of Dirichlet problem II. $L^p$ estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Hayk Aleksanyan, Henrik Shahgholian, Per Sj\\\"olin","submitted_at":"2012-09-03T20:45:49Z","abstract_excerpt":"Let $u_\\e$ be a solution to the system $$ \\mathrm{div}(A_\\e(x) \\nabla u_{\\e}(x))=0 \\text{\\ in} D, \\qquad u_{\\e}(x)=g(x,x/\\e) \\text{\\ on}\\partial D, $$ where $D \\subset \\R^d $ ($d \\geq 2$), is a smooth uniformly convex domain, and $g$ is 1-periodic in its second variable, and both $A_\\e$ and $g$ reasonably smooth. Our results in this paper are two folds. First we prove $L^p$ convergence results for solutions of the above system, for non-oscillating operator, $A_\\e(x) =A(x)$, with the following convergence rate for all $1\\leq p <\\infty$ $$ \\|u_\\e - u_0\\|_{L^p(D)} \\leq C_p \\begin{cases} \\e^{1/2p}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.0483","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"}