I hope this is easy. Let $\Sigma \subset \mathbb{R}^n$ be bounded with smooth boundary (it could be the shell region between two balls). Assume moreover that $u$ belongs to the Sobolev space $H^1(\mathbb{R}^n) = W^{1,2}(\mathbb{R}^n)$ and $a:\mathbb{R}^n\to \mathbb{R}_{\ge 0}$ is a smooth bounded and non-negative function.
My question:
Is there any way of estimating the surface integral
$$\int \limits _{\partial \Sigma } a(x)|u(x)|^2\ dS_x$$
by some non-surface integral of these quantities that does not involve putting any derivative on $a$? One derivative on $u$ would be fine.
So for instance, if one could estimate it by $C\|\sqrt{a}u\|_{L^2}^2$ for some constant $C>0$, that would be terrific, but other estimates could also do.
What comes to my mind is some Green formula, or possibly the trace theorem, but I didn't succeed yet.
