Dirichlet Problem for Second-Order Abstract Differential Equations

We study the well-posedness in the space of continuous functions of the Dirichlet boundary value problem for a homogeneous linear second-order differential equation u''+Au = 0, where A is a linear closed densely defined operator in a Banach space. We give necessary conditions for the well-posedness, in terms of the resolvent operator of A. In particular we obtain an estimate on the norm of the resolvent at the points k^2, where k is a positive integer, and we show that this estimate is the best possible one, but it is not sufficient for the well-posedness of the problem. Moreover we characterize the bounded operators for which the problem is well-posed.