schauder estimates of PDE with constant coefficient

Question

In the highlighted part, I dont understand what are these steps doing:

Why is $$\int A \, \nabla{w} \nabla{w} = \int F(x) \nabla{w}$$

and

$$\int F(x) \nabla{w}=\int (F(x)-F_R) \nabla {w}$$

and why $\nabla{v} \in H^1$?

THANKS!!

Answer 1

So we know that $\operatorname{div}(A\nabla w)=\operatorname{div}(F)$. Now,

$$\int_{B_R}A\nabla w\nabla w\,dx=-\int_{B_R}w\,\operatorname{div}(A\nabla w)\,dx=-\int_{B_R}w\,\operatorname{div}(F)\,dx=\int_{B_R}F\cdot\nabla w\,dx.$$

Perhaps you could say what $F_R$ is? in this case it seems it must be zero, is it the mean integral of $F$ perhaps?

$\nabla v$ is in $H^1$ because $v$ is in $H^2$, which means any second order weak derivative of $v$ is in $L^2$, (and lower order), so that means any first order weak derivative of $\nabla v$ is in $L^2$ (as well as $\nabla v$ itself), all in all this means $\nabla v\in H^2$.