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 Trudy Mat. Inst. Steklova, 2019, Volume 306, Pages 56–74 (Mi tm3997)

On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation

A. K. Gushchin

Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia

Abstract: The behavior of solutions of a second-order elliptic equation near a distinguished piece of the boundary is studied. On the remaining part of the boundary, the solutions are assumed to satisfy the homogeneous Dirichlet conditions. A necessary and sufficient condition is established for the existence of an $L_2$ boundary value on the distinguished part of the boundary. Under the conditions of this criterion, estimates for the nontangential maximal function of the solution hold, the solution belongs to the space of $(n-1)$-dimensionally continuous functions, and the boundary value is taken in a much stronger sense.

DOI: https://doi.org/10.4213/tm3997

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English version:
Proceedings of the Steklov Institute of Mathematics, 2019, 306, 47–65

Bibliographic databases:

UDC: 517.956.223
Revised: October 10, 2018
Accepted: May 30, 2019

Citation: A. K. Gushchin, “On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation”, Mathematical physics and applications, Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov, Trudy Mat. Inst. Steklova, 306, Steklov Math. Inst. RAS, Moscow, 2019, 56–74; Proc. Steklov Inst. Math., 306 (2019), 47–65

Citation in format AMSBIB
\Bibitem{Gus19} \by A.~K.~Gushchin \paper On the Existence of $L_2$ Boundary Values of Solutions to an Elliptic Equation \inbook Mathematical physics and applications \bookinfo Collected papers. In commemoration of the 95th anniversary of Academician Vasilii Sergeevich Vladimirov \serial Trudy Mat. Inst. Steklova \yr 2019 \vol 306 \pages 56--74 \publ Steklov Math. Inst. RAS \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm3997} \crossref{https://doi.org/10.4213/tm3997} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=4040765} \elib{https://elibrary.ru/item.asp?id=43226277} \transl \jour Proc. Steklov Inst. Math. \yr 2019 \vol 306 \pages 47--65 \crossref{https://doi.org/10.1134/S0081543819050067} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000511670100006} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85077354391} 

• http://mi.mathnet.ru/eng/tm3997
• https://doi.org/10.4213/tm3997
• http://mi.mathnet.ru/eng/tm/v306/p56

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This publication is cited in the following articles:
1. A. K. Gushchin, “The boundary values of solutions of an elliptic equation”, Sb. Math., 210:12 (2019), 1724–1752
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