Noncoercive boundary value problems for the Laplace equation with a spectral parameter

Abstract

In this paper we find conditions that guarantee that irregular boundary value problems for elliptic differential-operator equations of the second order in an interval are coercive with a defect and fredholm; compactness of a resolvent and estimations by spectral parameter; completeness of root functions. We apply this result to find some algebraic conditions that guarantee that irregular boundary value problems for elliptic partial differential equations of the second order in cylindrical domains have the same properties. Apparently this is the first paper where the regularity of an elliptic boundary value problem is not satisfied on a manifold of the dimension equal to the dimension of the boundary. Nevertheless, the problem is fredholm and the resolvent is compact. It is interesting to note that the considered boundary value problems for elliptic equations in a cylinder being with separating variables are noncoercive.