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 Sibirsk. Mat. Zh., 2017, Volume 58, Number 4, Pages 870–884 (Mi smj2905)

Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions

A. N. Polkovnikov, A. A. Shlapunov

Siberian Federal University, Krasnoyarsk, Russia

Abstract: Let $D$ be an open connected subset of the complex plane $\mathbb C$ with sufficiently smooth boundary $\partial D$. Perturbing the Cauchy problem for the Cauchy–Riemann system $\bar\partial u=f$ in $D$ with boundary data on a closed subset $S\subset\partial D$, we obtain a family of mixed problems of the Zaremba-type for the Laplace equation depending on a small parameter $\varepsilon\in(0,1]$ in the boundary condition. Despite the fact that the mixed problems include noncoercive boundary conditions on $\partial D\setminus S$, each of them has a unique solution in some appropriate Hilbert space $H^+(D)$ densely embedded in the Lebesgue space $L^2(\partial D)$ and the Sobolev–Slobodetskiĭ space $H^{1/2-\delta}(D)$ for every $\delta>0$. The corresponding family of the solutions $\{u_\varepsilon\}$ converges to a solution to the Cauchy problem in $H^+(D)$ (if the latter exists). Moreover, the existence of a solution to the Cauchy problem in $H^+(D)$ is equivalent to boundedness of the family $\{u_\varepsilon\}$ in this space. Thus, we propose solvability conditions for the Cauchy problem and an effective method of constructing a solution in the form of Carleman-type formulas.

Keywords: Cauchy–Riemann operator, Cauchy problem, Zaremba problem, small parameter, Laplace equation.

 Funding Agency Grant Number Ministry of Education and Science of the Russian Federation 14.Y26.31.0006ÍØ-9149.2016.1 The authors were supported by the Government of the Russian Federation (Grant 14.Y26.31.0006) and the State Maintenance Program for Leading Scientific Schools (Grant NSh-9149.2016.1).

DOI: https://doi.org/10.17377/smzh.2017.58.414

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English version:
Siberian Mathematical Journal, 2017, 58:4, 676–686

Bibliographic databases:

UDC: 517.35+517.53
MSC: 35R30

Citation: A. N. Polkovnikov, A. A. Shlapunov, “Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions”, Sibirsk. Mat. Zh., 58:4 (2017), 870–884; Siberian Math. J., 58:4 (2017), 676–686

Citation in format AMSBIB
\Bibitem{PolShl17} \by A.~N.~Polkovnikov, A.~A.~Shlapunov \paper Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions \jour Sibirsk. Mat. Zh. \yr 2017 \vol 58 \issue 4 \pages 870--884 \mathnet{http://mi.mathnet.ru/smj2905} \crossref{https://doi.org/10.17377/smzh.2017.58.414} \elib{https://elibrary.ru/item.asp?id=29947457} \transl \jour Siberian Math. J. \yr 2017 \vol 58 \issue 4 \pages 676--686 \crossref{https://doi.org/10.1134/S0037446617040140} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000408727100014} \elib{https://elibrary.ru/item.asp?id=31088397} \scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85028544989} 

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This publication is cited in the following articles:
1. Anastasiya S. Peicheva, “Regularization of the Cauchy problem for elliptic operators”, Zhurn. SFU. Ser. Matem. i fiz., 11:2 (2018), 191–193
2. Yu. Shefer, A. Shlapunov, “On regularization of the Cauchy problem for elliptic systems in weighted Sobolev spaces”, J. Inverse Ill-Posed Probl., 27:6 (2019), 815–834
3. Alexander N. Polkovnikov, “On initial boundary value problem for parabolic differential operator with non-coercive boundary conditions”, Zhurn. SFU. Ser. Matem. i fiz., 13:5 (2020), 547–558
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