Exercise about a functional and sobolev spaces

Question

I have to solve the following exercise. Given the functional $$ I(w, \Omega)=\int_{\Omega}F(Dw(x))dx $$ where $F\in C^2(\mathbb{R}^n)$ satisfies $$ F(p)\geq\alpha|p|^q-\beta $$ for some $\alpha>0$, $\beta\geq0$ and $n<q<+\infty$ and $\Omega$ is a bounded domain, I have to prove that if $u$ is a minimizer of $I$ in the space $$ W^{1, q}_g(\Omega)=\{w\in W^{1, q}(\Omega): w-g\in W^{1, q}_0(\Omega)\},\quad g\in W^{1, q}_{loc}(\Omega)\cap C^0(\overline{\Omega}) $$ (where $W^{1, q}_{loc}:=\{u\in L^q_{loc}(\Omega)\ |\ \ \exists \ Du=(D_1u\ldots D_nu):D_iu\in L^{q}_{loc}(\Omega)\}$), that is $I(u, \Omega)\leq I(u+\psi, \Omega)$, for all $\psi\in W^{1, q}_0(\Omega)$, then $I(u, \Omega')\leq I(u+\psi, \Omega')$, $u\in W^{1, q}_{loc}(\Omega)$, for all $\psi\in W^{1, q}_0(\Omega')$, $\Omega'\subset\subset\Omega$. Some hints? Thank You

Answer 1

Any $\psi\in W^{1, q}_0(\Omega')$ can be extended by zero to a function $\bar{\psi} \in W^{1, q}_0(\Omega)$, so you can use the assumption and subtract the integral over $\Omega \setminus \Omega '$ from both sides.