Boundedness of the derivative of the trace of an H^1 function As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter.
Suppose $u_n\in H^1(B_1)$ (actually, $u_n$ are smooth), where $B_1$ is the unit ball in $\mathbb{R}^N$, and that we know $||u_n||_{H^1} \leq C$, so that up to a subsequence, $u_n \rightharpoonup u$ in $H^1(B_1)$ weakly.  What can we say about the boundedness of the quantity
$\int_{\partial B_1} \nabla u_n(x) \cdot n(x)\;d\mathcal{H}^{N-1}(x)$,
where $n(x)$ is the unit normal to $\partial B_1$?
In particular, is this quantity always finite under these hypotheses, or is there a counterexample that shows this blows up for a bounded subset of $H^1$?
 A: If you restrict a $H^s$-function $f$ to a submanifold of codimension $k$ you get a $H^{s-k/2}$ function. 
EDIT: But only if $s$ is large enough (Sobolev lemma threshold). Thus the rest of my answer does not work, and I deleted it. 
A: Clearly, what you call $\newcommand{\bn}{\boldsymbol{n}}$ $\nabla u\cdot \bn $ is the  normal derivative $\frac{\partial u}{\partial \bn}$. The trace theorem (see e.g. Lions  and Magenes, Non-Homogeneous Boundary Value Problems and Applications. I, Thm. 9.4, Chap 1) shows that for  $s> \frac{3}{2}$  the restriction map $\newcommand{\pa}{\partial}$
$$R: C^\infty(B_1)\ni u\mapsto (u|_{\pa B_1}, \frac{\pa u}{\pa \bn})\in C^\infty(\pa B_1)\times C^\infty(\pa B_1) $$
extends to a continuous  split surjective map
$$ H^s(B_1)\to H^{s-\frac{1}{2}}(\pa B_1)  \times H^{s-\frac{3}{2}}(\pa B_1). $$
The result is optimal because Theorem  9.5, Chap. 1 op. cit.  shows that if $s\leq  \frac{3}{2}$, then for any $\phi\in C^\infty(\pa B_1)$ the linear functional
$$ C^\infty( B_1)\ni u\mapsto \int_{\pa B_1} \frac{\pa u (x)}{\pa \bn} \phi (x) dA(x)\in\mathbb{R} $$
is not continuous in the topology induced by $H^s(B_1)$. In particular, if $s\leq \frac{3}{2}$ there cannot exist a constant $C>0$ such that
$$ \left|\int_{\pa B_1} \frac{\pa u}{\pa \bn} dA\right| \leq C\Vert u\Vert_{H^s(B_1)},\;\;\forall  u\in C^\infty(B_1), $$
so that there exists a sequence $u_k\in C^\infty(B_1)$ such that $\Vert u_k\Vert_{H^1(B_1)}\leq 1$  and
$$  \left|\int_{\pa B_1} \frac{\pa u_k}{\pa \bn} dA\right| \to \infty. $$
A: The second order term in your PDE is the Laplacian. Even though in $H^1$ you cannot define the trace on the boundary, the Laplacian helps. I am not sure of which particular problem you are thinking of (many things are called Ginzburg-Landau), but the Pohozaev identity usually helps. See for example 'Vortices in the Magnetic Ginzburg-Landau Model' Here by Sandier and Serfaty -where many references of its use are given. 
They show for example that for
$$
-\Delta u = \frac{u}{\epsilon^2}(1-|u|^2) \textrm{ in }\Omega
$$ 
with $u=g$ on $\partial\Omega$, $|g|=1$ and $\Omega$ star-shaped, there holds (Lemma 5.2)
$$
\int_\Omega \frac{(1-|u|^2)^2}{\epsilon^2} + \int_{\partial \Omega} \left|\frac{\partial u}{\partial n}\right|^2 \leq C(\Omega,g),
$$ 
So the normal derivative is $L^2(\partial\Omega)$, and this does not follow from $u\in H^1(\Omega)$. 
The Pohozaev identity is to use as a test function $x\cdot \nabla u$ in the case of a ball, that is, a test function that will equal $\partial_n u$ on the boundary. This idea is also known under the name of Morawetz multiplier, and Rellich identity. 
