A non-autonomous variational problem describing a nonlinear Timoshenko beam

We study the non-autonomous variational problem: \begin{equation*} \inf_{(\phi,\theta)} \bigg\{\int_0^1 \bigg(\frac{k}{2}\phi'^2 + \frac{(\phi-\theta)^2}{2}-V(x,\theta)\bigg)\text{d}x\bigg\} \end{equation*} where $k>0$, $V$ is a bounded continuous function, $(\phi,\theta)\in H^1([0,1])\times L^2([0,1])$ 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, when studying the equilibria of a nonlinear Timoshenko beam under distributed load, and in classical dynamics of coupled particles in time-depending 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.