Let $u$ be a function in the Sobolev space $W^{\frac 3 2, 2}(\mathbb R^n)$. Let $\Sigma\subset \mathbb R^n$ be the graph of a Lipschitz function $f:B_{n-1}(0,1) \subset \mathbb R^{n-1} \rightarrow \mathbb R$ (where $B_{n-1}$ is the unit ball in $\mathbb R^{n-1}$).
By some trace theorem in Sobolev spaces, we have $\nabla u|_{\Sigma} \in (L^2(\Sigma))^n$.
Consider a sequence $(a_i)_i$ of positive real numbers with $\lim_{i \rightarrow \infty} a_i = 0$ and the graphs $\Sigma_i$ of the functions $f + a_i$ for $i \in \mathbb N$. Thus, the graph $\Sigma_i$ is just the graph $\Sigma$ displaced.
Is it true that $\nabla u|_{\Sigma_i} \rightarrow \nabla u|_\Sigma$ in $L^2(\Sigma)^n$? Is there any reference for this kind of result (I don't know how it would be called in the literature)?
