Inequalities for solutions of mixed boundary value problems for elastic plates

In two recent papers [2,3] I the authors prese nted methods for obtaining pointwise bounds in the three most common boundary value problems for elasti c plates . These bounds were of a priori type, that is they held for a class of functions r equired to sati sfy only smoothness conditions . Hence one could approximate the (unknown) solution of one of these proble ms in terms of essentially arbitrary functions, and the inequaliti es gave bounds on the error. In this paper we derive s imilar a priori bounds in the three most common mixed boundary value problems for elastic plates . For simplicity we consider only the case of a simply conn ec ted region R whose boundary I consists of two disjoint portions II, and I 2 (each co nnected) on which different se ts of boundary conditions are imposed. It will be clear how the results are to be extended if II, and/or I 2 are not connec ted or if R is multiply connected. In this paper we shall res tric t our atte ntion to the proble m of obtaining bounds for the L2 integrals of an arbitrary sufficiently smooth function w in term s of L2 integrals of quantities which are data whenever the arbitrary fun c tion w is actually the solution u to the problem in ques tion. By use of mean value inequalities and the Rayleigh-Ritz technique as indicated in [2,3], the desired pointwise bounds are then obtained. The well known Rayleigh-Ritz tec hniqu e consis ts in choosing tV w = U L a;