Let $\Omega\subset \Bbb R^n$ be a $C^{2}$ domain (open and bounded) and let $p\in(1,\infty)$. Suppose $u\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$ is a weak solution to the fourth-order elliptic system $$ \Delta^2 u = -\text{div}\, F $$ in the sense that for any $\varphi\in C^\infty_c(\Omega;\Bbb R^m)$, we have $$ \int_{\Omega}\sum_{i,\alpha,\beta} (\partial_{\alpha} \partial_{\beta}u^i) (\partial_{\alpha} \partial_{\beta}\varphi^i) \,dx = \int_{\Omega}\sum_{i,\alpha} F^i_\alpha (\partial_{\alpha}\varphi^i) \,dx. $$ Here $F$ is only assumed to be in $L^{q}(\Omega;\Bbb R^{m\times n})$, where $1/p+1/q=1$.
Naively, it appears to me that we can expect $u$ to be of class $W^{3,r}_{loc}$ for some suitable $r$, at least in the interior of $\Omega$. As for the regularity up to boundary, I am even less sure about what kind of constraints are needed to guarantee sufficient regularity.
Q: What are some papers or monographs that cover this kind of topic?
For example, what kind of boundary regularity can we expect from assuming that $u|_{\partial \Omega} = \text{Tr}(w)$ for some $w\in W^{1,p}\cap W^{2,2}(\Omega;\Bbb R^m)$? Would assuming that $w\in C^1(\bar{\Omega};\Bbb R^m)$ be enough to conclude that $u \in C^1(\bar{\Omega};\Bbb R^m)$? Do we need to assume more regularity on $\partial \Omega$ or can we even drop it down to the class $C^{1,1}$?
Note that I am only interested in the regularity aspect of such a solution $u$. I won't mind if the reference doesn't mention the existence theory.
