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 $H^1$?



Boundedness of the derivative of the trace of an H^1 function
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


$$ \left|\int\_{\pa B\_1} \frac{\pa u\_k}{\pa \bn} dA\right| \to \infty. $$


