My question is inspired by the work of Lamm and Rivière : Conservation Laws for Fourth Order Systems in Four Dimensions
Here is the setting of the problem. Let $u\in W^{2,2}(B(0,1),S^n)$, where $B(0,1)$ is the unit ball of $\mathbb{R}^4$ and for a fixed $n\geq 1$. We define the bi-energy $B$ as: $$B(u):=\int_{B(0,1)} \vert \Delta u\vert^2 \, dx $$
(or $B(u):=\int_{B(0,1)} \Vert (\Delta u)^T \Vert^2 \, dx $ , where $(\Delta u)^T$ is the projection of $\Delta u$ on $T_u S^n$).
In fact it is quite important to notice that
$$\Vert (\Delta u)^T \Vert^2= \Vert \Delta u\Vert^2 -\Vert \nabla u\Vert^4 \; (1)$$
Critical points of this equation are known as biharmonic maps, generalizing harmonic in dimension $4$. The Euler-Lagrange equation is quite complicated see the introduction of [1], but it is mostly like $$\Delta^2 u= \langle \nabla u, \nabla \Delta u\rangle + \vert \nabla^2 u\vert^2 + \vert\nabla u\vert^2 \Delta u +\vert \nabla u\vert^4 $$
In the article they prove an $\epsilon$-regularity result, that is to say if the $W^{2,2}$-norm of $u$ is small enough (in fact in the sphere case $B(u)$ is enough because of the remark above) then you control all the derivatives on $B(0,1/2)$. It is a very powerful result. But I wonder about a global one. Of course you can't expect to control anything more than $W^{2,2}$ on $B(0,1)$ but at least can you prove that $\Delta^2u$ is in $L^1$?
Here is the precise question, does there exist $\epsilon >0$ such that if $\phi\in W^{2,2}$ with $B(\phi)\leq \epsilon$ and $u\in W^{2,2}(B(0,1),S^n)$ is a minimizer of $B$ under the boundary condition $u=\phi,\partial_\nu u=\partial_\nu \phi$ on $\partial B(0,1)$, then $$\Delta^2 u \in L^1$$ or even $$\Vert \Delta u\Vert_1\leq f(\epsilon)$$ with some $f\in C(\mathbb{R}^+)$ with $f(0)=0$.
It would be very surprising to me if not since it is for free for harmonic maps since the equation is $\Delta u =\vert \nabla u\vert^2$, hence the $L^1$-norm of the Laplacian is automatically controlled by the energy.
