We consider $\n$-hypersurfaces $\Sij$ with interior $\Ej$ whose mean curvature are given by the trace of an ambient Sobolev function $\uj \in W^{1,\p}(\rel^{\n+1})$

        
(0.1)   \mean_\Sij = u_j\nuEj
        \quad \mbox{on } \Sij,

where $\nuEj$ denotes the inner normal of $\Sij$. We investigate (0.1) when $\Sij \rightarrow \Si$ weakly as varifolds and prove that $\Si$ is an integral $\n$-varifold with bounded first variation which still satisfies (0.1) for $\uj \rightarrow \u,\Ej \rightarrow \E$. $\p$ has to satisfy \begin{displaymath} \p > \frac{1}{2} (\n + 1) \end{displaymath} and $\p \geq \frac{4}{3} \mbox{ if } \n = 1$. The difficulty is that in the limit several layers can meet at $\Si$ which creates cancellations of the mean curvature.