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This article is cited in 5 scientific papers (total in 5 papers)
On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation
S. V. Gaidenko
Abstract:
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.
Received: 13.06.1979
Citation:
S. V. Gaidenko, “On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a second order elliptic equation”, Math. USSR-Sb., 39:1 (1981), 107–123
Linking options:
https://www.mathnet.ru/eng/sm2494https://doi.org/10.1070/SM1981v039n01ABEH001475 https://www.mathnet.ru/eng/sm/v153/i1/p116
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