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Ufa Mathematical Journal, 2022, Volume 14, Issue 4, Pages 14–25
DOI: https://doi.org/10.13108/2022-14-4-14
(Mi ufa632)
 

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

On energy functionals for second order elliptic systems with constant coefficients

A. O. Bagapshabc, K. Yu. Fedorovskiydce

a Federal Research Center “Informatics and Control”, RAS, Vavilova str. 44, bld. 2, 119333, Moscow, Russia
b Bauman Moscow State Technical University, 2nd Baumanskaya str 5, bld. 1, 105005, Moscow, Russia
c Moscow Center of Fundamental and Applied Mathematics, Lomonosov Moscow State University, Leninskie gory, 1, 119991, Moscow, Russia
d Saint-Petersburg State University, 14 line Vasilievsky island, 29b, 199178, Saint-Petersburg, Russia
e Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Leninskie gory, 1, 119991, Moscow, Russia
References:
Abstract: We consider the Dirichlet problem for second-order elliptic systems with constant coefficients. We prove that non-separable strongly elliptic systems of this type admit no nonnegative definite energy functionals of the form
$$ f\mapsto\int\limits_{D}\varPhi(u_x,v_x,u_y,v_y)\,dxdy, $$
where $D$ is the domain in which the problem is considered, $\varPhi$ is some quadratic form in $\mathbb{R}^4$ and $f=u+iv$ is a function of the complex variable. The proof is based on reducing the considered system to a special (canonical) form when the differential operator defining this system is represented as a perturbation of the Laplace operator with respect to two small real parameters, the canonical parameters of the considered system. In particular, the obtained result show that it is not possible to extend the classical Lebesgue theorem on the regularity of an arbitrary bounded simply connected domain in the complex plane with respect to the Dirichlet problem for harmonic functions to strongly elliptic second order equations with constant complex coefficients of a general form is not possible. This clarifies a number of difficulties arising in this problem, which is quite important for the theory of approximations by analytic functions.
Keywords: second order elliptic system, canonical representation of second order elliptic system, Dirichlet problem, energy functional.
Funding agency Grant number
Foundation for the Development of Theoretical Physics and Mathematics BASIS
Ministry of Science and Higher Education of the Russian Federation 0705-2020-0047
075-15-2021-602
The work is supported by the Foundation for Developing Theoretical Physics and Mathematics “Basis” and by the Ministry of Science and Higher Education of Russian Federation within the project 0705-2020-0047. Lemmata 3.1 and 3.2 are obtained under the grant of the Government of Russian Federation for state support of scientific researches under supervision of leading scientists, agreement 075-15-2021-602.
Received: 01.10.2022
Bibliographic databases:
Document Type: Article
UDC: 517.53+517.95
MSC: 30E25, 35J25
Language: English
Original paper language: Russian
Citation: A. O. Bagapsh, K. Yu. Fedorovskiy, “On energy functionals for second order elliptic systems with constant coefficients”, Ufa Math. J., 14:4 (2022), 14–25
Citation in format AMSBIB
\Bibitem{BagFed22}
\by A.~O.~Bagapsh, K.~Yu.~Fedorovskiy
\paper On energy functionals for second order elliptic systems with constant coefficients
\jour Ufa Math. J.
\yr 2022
\vol 14
\issue 4
\pages 14--25
\mathnet{http://mi.mathnet.ru//eng/ufa632}
\crossref{https://doi.org/10.13108/2022-14-4-14}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4516556}
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  • https://www.mathnet.ru/eng/ufa632
  • https://doi.org/10.13108/2022-14-4-14
  • https://www.mathnet.ru/eng/ufa/v14/i4/p16
  • This publication is cited in the following 3 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Уфимский математический журнал
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    Russian version PDF:19
    English version PDF:25
    References:21
     
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