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 Mat. Sb. (N.S.), 1980, Volume 111(153), Number 1, Pages 116–134 (Mi msb2494)

This article is cited in 4 scientific papers (total in 4 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.

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English version:
Mathematics of the USSR-Sbornik, 1981, 39:1, 107–123

Bibliographic databases:

UDC: 517.946
MSC: Primary 35J67; Secondary 35J25
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”, Mat. Sb. (N.S.), 111(153):1 (1980), 116–134; Math. USSR-Sb., 39:1 (1981), 107–123

Citation in format AMSBIB
\Bibitem{Gai80} \by S.~V.~Gaidenko \paper On exceptional sets on the boundary and the uniqueness of solutions of the Dirichlet problem for a~second order elliptic equation \jour Mat. Sb. (N.S.) \yr 1980 \vol 111(153) \issue 1 \pages 116--134 \mathnet{http://mi.mathnet.ru/msb2494} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=560466} \zmath{https://zbmath.org/?q=an:0462.35026|0429.35029} \transl \jour Math. USSR-Sb. \yr 1981 \vol 39 \issue 1 \pages 107--123 \crossref{https://doi.org/10.1070/SM1981v039n01ABEH001475} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1981LQ97300005} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. S. V. Gaidenko, “On exceptional sets on the boundary and uniqueness of a solution of the second and third boundary value problems for an elliptic equation”, Math. USSR-Sb., 53:2 (1986), 351–366
2. Gaidenko S., “Growth of Harmonic-Functions Near a Boundary”, Differ. Equ., 21:5 (1985), 557–561
3. A. K. Gushchin, “On the Dirichlet problem for a second-order elliptic equation”, Math. USSR-Sb., 65:1 (1990), 19–66
4. Chabrowski J., “The Dirichlet Problem with l(2)-Boundary Data for Elliptic Linear-Equations”, Lect. Notes Math., 1482 (1991), 1–171
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