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

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.