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 J. Comp. Eng. Math., 2017, Volume 4, Issue 3, Pages 27–34 (Mi jcem97)

Computational Mathematics

Solvability of the Showalter–Sidorov problem for Sobolev type equations with operators in the form of first-order polynomials from the Laplace–Beltrami operator on differential forms

D. E. Shafranov, N. V. Adukova

South Ural State University, Chelyabinsk, Russian Federation

Abstract: We consider solvability of the Showalter–Sidorov problem for the Barenblatt–Zheltov–Kochina equations and the Hoff linear equation. The equations are linear representatives of the class of linear Sobolev type equations with an irreversible operator under derivative. We search for a solution to the problem in the space of differential $k$-forms defined on a Riemannian manifold without boundary. Both equations are the special cases of an equation with operators in the form of polynomials of the first degree from the Laplace–Beltrami operator, generalizing the Laplace operator in spaces of differential $k$-forms up to a sign. Applying the Sviridyuk theory and the Hodge-Kodaira theorem, we prove an existence of the subspace in which there exists a unique solution to the problem.

Keywords: Sobolev type equation, Riemannian manifolds, manifold without boundary, differential forms, Laplace–Beltrami operator.

DOI: https://doi.org/10.14529/jcem170304

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UDC: 517.9
MSC: 57R35
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Citation: D. E. Shafranov, N. V. Adukova, “Solvability of the Showalter–Sidorov problem for Sobolev type equations with operators in the form of first-order polynomials from the Laplace–Beltrami operator on differential forms”, J. Comp. Eng. Math., 4:3 (2017), 27–34

Citation in format AMSBIB
\Bibitem{ShaAdu17} \by D.~E.~Shafranov, N.~V.~Adukova \paper Solvability of the Showalter--Sidorov problem for Sobolev type equations with operators in the form of first-order polynomials from the Laplace--Beltrami operator on differential forms \jour J. Comp. Eng. Math. \yr 2017 \vol 4 \issue 3 \pages 27--34 \mathnet{http://mi.mathnet.ru/jcem97} \crossref{https://doi.org/10.14529/jcem170304} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3708702} \elib{http://elibrary.ru/item.asp?id=30289925}