Consider the energy functional $e(\cdot)$ \begin{align*} e(f,Q)&=\int_a^b \bigg\{f^4\bigg[1+\|\frac{d}{dr}Q\|^2+f^2\dot f^2\bigg]\bigg\} \,dr, \end{align*} over the space of \begin{equation*} {\mathcal E}:=\left\{ (Q, f) : \begin{array}{l} Q \in W^{1,4}([a,b],{\bf SO}(n)),\\ Q(a)=Q(b)=I,\\ f \in W^{1,4}[a,b],\\ \dot f>0 \mbox{ ${\cal L}^1$-a.e. on $(a,b)$},\\ f(a)=a, f(b)=b. \end{array} \right\} \end{equation*} It is easy to show that, energy functional $e(\cdot)$ is coercive when $Q\in W^{1,2}$ and $f\in W^{1,2}$, in another words, there exists $d=d(n, a, b)>0$ such that \begin{equation*} e(f, Q) \ge d ( \|Q\|^2_{W^{1,2}} + \|f\|^2_{W^{1,2}}). \end{equation*} Now my question is: The energy functional $e(\cdot)$ is coercive on the space $\mathcal E$ or not? In another words could we find $\gamma=\gamma(n, a, b)>0$ such that \begin{equation*} e(f, Q) \ge \gamma ( \|Q\|^4_{W^{1,4}} + \|f\|^4_{W^{1,4}}). \end{equation*} for all $(f, Q)\in \mathcal E$.