On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation

The Dirichlet problem is considered for a linear elliptic equation of second order in $n$-dimensional domain $Q$, $n\geqslant2$, with smooth boundary $\partial Q$ in the case where the generalized solution of this equation takes boundary values everywhere on the boundary but an exceptional set $\mathscr E\subset\partial Q$. It is proved that for $n/(n-1)\leqslant p<\infty$ the space $L_p(Q)$ is a class of uniqueness for such a problem if $\mathscr E$ has finite Hausdorff measure of order $n-q$, where $\frac1p+\frac1q=1$. By an example of the Dirichlet problem for Laplace's equation it is shown that the indicated order of the Hausdorff measure is best possible.
Bibliography: 14 titles.