I have already posted this question on Math StackExchange. Suppose you have a sequence of $C^1$ functions $\{\phi_n\}_{n\in \mathbb{N}}$ defined on $\mathbb{R}^n$ that converges in $C^{1}_{\mathrm{loc}}$ to a function $\phi$, as $n \to +\infty$. By $C^1_{\mathrm{loc}}$ convergence I mean that $$ \lim_{n\to +\infty} \sup_{x\in K} |\phi_n(x)-\phi(x)|+|\nabla \phi_n(x)- \nabla \phi(x)|=0 $$ for every $K\Subset \mathbb{R}^n$. Suppose also that all functions are coercive ($\{\phi(x) \leq M\}$ and $\{\phi_n(x) \leq M\}$ are compact for all $n\in \mathbb{N}$ and $M \in \mathbb{R}$). Let $a<b$ be in the image of $\phi$ and $\phi_n$ for every $n\in \mathbb{N}$. Can one show that $$ \lim_{n \to +\infty} \mathcal{H}^{n-1}(\partial \{\phi_n(x) \leq t\} )= \mathcal{H}^{n-1}(\partial \{\phi(x)\leq t\}) $$ for almost every $t \in [a,b]$?
The liminf identity $$ \liminf_{n \to +\infty} \mathcal{H}^{n-1}(\partial \{\phi_n(x) \leq t\} )= \mathcal{H}^{n-1}(\partial \{\phi(x)\leq t\}) $$ follows by Fatou's lemma and coarea formula. Indeed, the '$\geq$' inequality is a consequence of the lower semicontinuity of the perimeter under $L^1$ convergence of sublevel sets. On the other hand, \begin{align} 0 &\leq \int_a^b \liminf_{n\to +\infty}\mathcal{H}^{n-1}(\partial \{\phi_n(x) \leq t\}) - \mathcal{H}^{n-1}(\partial \{\phi(x)\leq t\}) \,\mathrm{d}t\\&\leq \liminf_{n\to+\infty}\int_a^b \mathcal{H}^{n-1}(\partial \{\phi_n(x) \leq t\} )- \mathcal{H}^{n-1}(\partial \{\phi(x)\leq t\}) \,\mathrm{d}t\\ &=\liminf_{n\to+\infty}\int\limits_{\{a\leq \phi_n\leq b\}} |\nabla \phi_n(x)| \,\mathrm{d} x - \int\limits_{\{a\leq \phi\leq b\}} |\nabla \phi(x)| \,\mathrm{d} x =0, \end{align} where the last identity is a consequence of the $C^1$ convergence. Can this liminf be improved to a full limit?
Thank you in advance.
