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
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.