Assume $\Omega\subset \mathbb R^N$ is open bounded, smooth boundary. Also assume $S\subset \Omega$ is a smooth hyper surface such that $0<\mathcal H^{N-1}(S)<+\infty$.
Now, given a positive function $u\in SBV(\Omega)$ and define function $\tau (x)=\operatorname{dist}(x,S)$ for $x\in\Omega$. My question: do we have the following equality hold? $$ \lim_{t\to 0}\left\{\sup_{s\in(t/2,t)}\left\{\int_{\{y:\tau(y)=s\}} u(y)\,d\mathcal H^{N-1} \right\}\right\}= \int_S(u^+(x)+u^-(x))d\mathcal H^{N-1} $$ The question is clear when $N=1$. But I got stuck on case $N>1$.
Any help is really welcome!
PS: the definition $$ u^+(x):=\inf\{s\in[-\infty,+\infty]:\,\,\lim_{r\to0^+}\frac{\mathcal{H}^N(\{y\in B(x,r):\,\, u(y)>s\})}{r^N}=0\}, $$ $$ u^-(x):=\sup\{s\in[-\infty,+\infty]:\,\,\lim_{r\to0^+}\frac{\mathcal{H}^N(\{y\in B(x,r):\,\, u(y)<s\})}{r^N}=0\}, $$
