Let $\Omega$ be a bounded open set with smooth boundary and $E$ a set of finite perimeter in $\Omega$, i.e. $$P(E;\Omega)=\left\{\int_E\text{div}\: T\:dx:T\in C^\infty_c(\Omega;\mathbb{R}^n), |T|\leq1\right\}<\infty.$$ According to De Giorgi's structure theorem the reduce boundary $\partial^\ast E$ is $(n-1)-$recitfiable; more precisely, there exist at most countably many $C^1$ surfaces $\{M_j\}_j$ and compact sets $\{K_j\}_j$ with $K_j\subset M_j$ and $\nu_E(x)^\perp = T_xM_j$ for all $x\in K_j$ such that $$\partial^\ast E = F\cup\bigcup_{j\in\mathbb{N}}K_j$$ and $\mathcal{H}^{n-1}(F)=0$.
Let $H\in W^{2,2}(\Omega)$, the Sobolev space of functions whose distributional second derivatives are functions in $L^2(\Omega)$. My question is the following: is the tangential gradient of $H$ (denoted $\nabla_E H$) a well defined object over $\partial^\ast E$? Since for a regular enough $H$ we have $$\nabla_EH=\nabla H -(\nu_E\cdot\nabla H)\nu_E\quad \text{on }\partial^\ast E$$ I believe that the question bolis down to being able to define $\nabla H$ on $\partial^\ast E$.
My intuition says this should be true, by I cannot find a way to adapt the proof for Lipschitz boundary to this case, I am not sure that the partition of unity is finite for any ball.
Any comment/suggestion on this issue would be appreciated.
