A non-autonomous variational problem describing a nonlinear timoshenko beam

We study the nonautonomous variational problem inf((Phi,theta)){integral(1)(0)(k/2 Phi '(2 )+ (Phi-theta)(2)/2 - V(x, theta))dx}, where k > 0, V is a bounded continuous function, (Phi, theta) is an element of H-1([0,1]) x L-2([0,1]), and Phi(0) = 0 . The peculiarity of the problem is its setting in the product of spaces of different regularity order. Problems with this form arise in elastostatics, in the study of equilibria of a nonlinear Timoshenko beam under distributed load, and in classical dynamics of coupled particles in time-dependent external fields. We prove the existence and qualitative properties of global minimizers and study, under additional assumptions on V , the existence and regularity of local minimizers.