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In this paper we take up the study of the Dirichlet problem for elliptic operators in upper-graph Lipschitz domains, with boundary values (taken in a pointwise nontangential sense) belonging to the Sarason space of functions with vanishing mean oscillations. We employ certain modified versions of boundary layer potentials in order to construct solutions satisfying certain vanishing Carleson measure conditions.
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Dorina Mitrea, Irina Mitrea, Marius Mitrea, The Dirichlet problem with VMO data in upper-graph Lipschitz domains. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 30 (2019), no. 4, pp. 701–732DOI 10.4171/RLM/868