Semigroup estimates and noncoercive boundary value problems

In this paper we find conditions that guarantee that irregular boundary value problems of the second order for elliptic differential-operator equations with a parameter in an interval are coercive with a defect. We also prove the compactness of the resolvent, estimates with respect to a spectral parameter and the completeness of the system of root functions. We apply these results to find some algebraic conditions that guarantee that irregular boundary value problems of the second order for elliptic partial differential equations with a parameter in cylindrical domains have the same properties. In this paper the regularity of an elliptic boundary value problem is not satisfied on a manifold of 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, though with separable variables, are noncoercive.