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

This article is cited in 3 scientific papers (total in 3 papers)

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.Y26.31.0006
НШ-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
Received: 25.10.2016

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
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\paper Construction of Carleman formulas by using mixed problems with parameter-dependent boundary conditions
\jour Sibirsk. Mat. Zh.
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\vol 58
\issue 4
\pages 870--884
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\crossref{https://doi.org/10.17377/smzh.2017.58.414}
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\transl
\jour Siberian Math. J.
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\vol 58
\issue 4
\pages 676--686
\crossref{https://doi.org/10.1134/S0037446617040140}
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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. Anastasiya S. Peicheva, “Regularization of the Cauchy problem for elliptic operators”, Zhurn. SFU. Ser. Matem. i fiz., 11:2 (2018), 191–193  mathnet  crossref
    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  crossref  mathscinet  zmath  isi  scopus
    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  mathnet  crossref
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