## Estimate a surface integral

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.

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Erm... I understand that you do not put the derivative on $a$, but your "terrific" bound doesn't put it on $u$ either, which is way too much to ask for in the case of a fixed region. Did you really mean $\nabla u$? – fedja Nov 25 2011 at 13:16
Thanks for the reply, I understand my question is rather vague, which I'm sorry for. But sure, I can do with less, for instance $C\|\sqrt{a}u\| \cdot \|\nabla u\|$. – Alex A Nov 25 2011 at 13:35
Well, a rather idiotic estimate is $\max|a|(\|u\|\|\nabla u\|)$. It formally satisfies your requirements but almost certainly is not what you want. On the other hand, all the estimates you suggested so far are patently impossible (imagine $a$ that is large on the boundary and almost zero away from it). Can we settle on something that is not obviously false but would still suffice for your purposes? – fedja Nov 25 2011 at 19:11
Sure, the problem is I would really need the factor/term involving $\|u\|$ to be of the form $\|au\|$. – Alex A Nov 30 2011 at 14:46