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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 theproduct of spaces of different regularity order. Problems with this form arisein elastostatics, when studying the equilibria of a nonlinear Timoshenko beamunder distributed load, and in classical dynamics  of coupled particles in time-depending external fields. We prove theexistence and qualitative properties of global minimizers and study, underadditional assumptions on $V$, the existence and regularity of localminimizers.

A non-autonomous variational problem describing a nonlinear Timoshenko beam