consider the subsequent pde (weak formulation):
$\int_\Omega D^m\phi:D^m\psi+ Df(D\phi):D\psi+(g h\circ\phi)\cdot\psi dx=0$.
In this case, $n\geq 2$, $\Omega=[0,1]^n$, $m>2+\frac{n}{2}$, $\phi\in H^m(\Omega,\Omega)\cap H^1_0(\Omega,\Omega)$ is the "solution" of the pde and a diffeomorphism, $\psi\in H^m(\Omega,\Omega)\cap H^1_0(\Omega,\Omega)$ is the space of test functions, $f\in C^\infty(\mathbb{R}^{n,n},\mathbb{R})$ is non-negative and polyconvex, $g\in H^1(\Omega,\mathbb{R})$ and $h\in L^2(\Omega,\Omega)$.
I could prove existence of solutions of this pde and interior regularity, i.e. $\phi\in H^{m+1}(\Omega')$ for every $\Omega'\Subset\Omega$.
My question: how can I prove boundary regularity, i.e. $\phi\in H^{m+1}(\Omega)\cap H^1_0(\Omega)$?
I read the books by Grisvard ("Elliptic Problems in Nonsmooth Domains") and some parts of the book by Gauge ("Elliptic Boundary Value Problems on Corner Domains"), but in the end I discovered that every time the assumptions in my case are too weak. Are techniques involving weighted Sobolev spaces appropriate in this case?
Thank you in advance!
