Gevrey's and trace regularity of a semigroup associated with beam equation and non-monotone boundary conditions

The main aim of the paper is to present a technique that allows to infer wellposedness, trace and Gevrey's regularity of hyperbolic-like PDE's with non-monotone boundary conditions. The lack of monotonicity prevents applicability of the known semigroup methods.In this paper we show how recently developed tools of microlocal analysis [D. Tataru, A priori estimates of Carleman's type in domains with boundary, J. Math. Pure Appl. 73 (1994) 353–387] combined with some spectral theory can be used successfully in order to obtain the needed inequalities. The method will be illustrated on a simple example of beam equation with non-monotone boundary conditions.