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 $\phi(0)=0$ in the sense of traces. 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.