Geodesic
$L_p$-estimates of the~solution of the~Dirichlet problem for second-order elliptic equations
Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 1-17

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The Dirichlet problem for second-order divergence-form elliptic equations with coefficients continuous in a closed domain is considered. A necessary and sufficient condition on the boundary of a compact domain ensuring the unique $L_p$-solubility of the problem in question and also a corresponding coercive $L_p$-estimate for all $p>1$ are obtained. 
@article{SM_1998_189_1_a0,
     author = {Yu. A. Alkhutov},
     title = {$L_p$-estimates of the~solution of {the~Dirichlet}  problem for second-order elliptic equations},
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     volume = {189},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_1_a0/}
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Yu. A. Alkhutov. $L_p$-estimates of the~solution of the~Dirichlet  problem for second-order elliptic equations. Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 1-17. http://geodesic.mathdoc.fr/item/SM_1998_189_1_a0/