Geodesic
The Dirichlet problem for a~second-order elliptic equation with an $L_p$ boundary function
Sbornik. Mathematics, Tome 203 (2012) no. 1, pp. 1-27

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We consider a Dirichlet problem in which the boundary value of a solution is understood as the $L_p$-limit, $p>1$, of traces of this solution on surfaces ‘parallel’ to the boundary. We suggest a setting of this problem which (in contrast to the notion of solution in $W_{p,\operatorname{loc}}^1$) enables us to study the solvability of the problem without making smoothness assumptions on the coefficients inside the domain. In particular, for an equation in selfadjoint form without lower-order terms, under the same conditions as those used for $p=2$, we prove unique solvability and establish a bound for an analogue of the area integral. Bibliography: 37 titles.
Keywords: Dirichlet problem, boundary value.
Mots-clés : elliptic equation 
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A. K. Gushchin. The Dirichlet problem for a~second-order elliptic equation with an $L_p$ boundary function. Sbornik. Mathematics, Tome 203 (2012) no. 1, pp. 1-27. http://geodesic.mathdoc.fr/item/SM_2012_203_1_a0/