How can I conclude this "gluing property" for these Sobolev functions?

Question

Let $u \in W^{1,2}(\Omega) \cap C(\Omega)$, where $\Omega$ is an open bounded domain in $R^n$ with smooth boundary.Let $B(x,R) \subset \overline{B(x,R)} \subset \Omega $ a ball. Consider $u^{\star} \in W^{1,2}B(x,R)$ satisfying $\Delta u^{\star} = 0$ in $B(x,R)$ and $u^{\star} - u \in W^{1,2}_0 (B(x,R))$.

Define a function $v $ in $\Omega$ by the equations $v(y) = u(y)$ in $\Omega - B(x,R)$ and $v(y) = \min (u(y), u^{\star} (y))$ in $B(x,R).$ I am reading a paper and the author says that $v \in W^{1,2}_{loc} (\Omega)$ . It is reasonable to expect this, but I dont know how to prove this.. Someone could point me a reference?

Answer 1

First notice that $v\in W^{1,2}(B)$ and then consider $u$ as a continuous extension of $v$ in $\Omega$.