The Dirichlet problem for a second-order elliptic equation with an $L_p$ boundary function

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.
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