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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. (English) Zbl 1458.76106

Summary: 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.

MSC:

76S05 Flows in porous media; filtration; seepage
35Q35 PDEs in connection with fluid mechanics
53Z05 Applications of differential geometry to physics
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References:

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