Geodesic
On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a~second order elliptic equation
Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 107-123

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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. 
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     author = {S. V. Gaidenko},
     title = {On exceptional sets on the boundary and the uniqueness of solutions of the {Dirichlet} problem for a~second order elliptic equation},
     journal = {Sbornik. Mathematics},
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     year = {1981},
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     url = {http://geodesic.mathdoc.fr/item/SM_1981_39_1_a4/}
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S. V. Gaidenko. On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a~second order elliptic equation. Sbornik. Mathematics, Tome 39 (1981) no. 1, pp. 107-123. http://geodesic.mathdoc.fr/item/SM_1981_39_1_a4/