Consider a positive definite locally Lipschitz function $V:\mathbb{R}^2\to\mathbb{R}_{\geq0}$.
Fix $c\in\mathbb{R}_{\geq0}$ and consider the sublevel-set $E_c=\{x:\in\mathbb{R}^2:V(x)\leq c\}$, define the function $v_c:\partial E_c\to\mathbb{R}^2$ given by \begin{equation*} v_c(x)=\dfrac{\nabla\chi_{E_c}(x)}{|\nabla\chi_{E_c}(x)|}, \end{equation*} where $\chi_{E_c}$ is the indicator function of $E_c$.[1]
I would like to define an indicator function $\xi$ for each sublevel set of $V$ whose gradient coincide the gradient of $V$. More precisely, for every $c\in\mathbb{R}_{\geq0}$, define a function $\xi_c:\mathbb{R}_{\geq0}\to[0,1]$ such that \begin{equation*} \xi_c(x)=\left\{\begin{array}{rclrcl} 1,&\text{if}&x&:V(x)\leq c\\ 0,&\text{if}&x&:V(x)\neq c, \end{array}\right. \end{equation*} and \begin{equation*} \nabla \xi_c(x)= \nabla V(x) \end{equation*}
I have no clue how to defined it. I would like to use it for sets of finite perimeter [1] satisfying Federer's definition of measure-theoretic normal to set [2].
[1] http://en.wikipedia.org/wiki/Caccioppoli_set#Notions_of_boundary
[2] Pfeffer, F. W. - The divergence theorem and sets of finite perimeter.
I think you misunderstand the basic concepts of the subject. The expression $\nabla \chi_{E_c}$ is not "a function $v:\partial E_c\to\mathbb R^2$", but a distribution. If your sublevel set $E_c$ happens to be a Caccioppoli set (which is not guaranteed; the sublevel set of a Lipschitz function can be any closed set) then $\nabla \chi_{E_c}$ is a vector-valued Radon measure supported on $\partial E_c$. It is still not a function; thus, it does not make sense to ask for it to agree with $\nabla V$.
Besides, $\nabla V$ itself is guaranteed to exist only a.e. (by Rademacher's theorem) and since $\partial E_c$ usually has zero Lebesgue measure, you may well have a situation where $\nabla V$ is not defined at any point of $\partial E_c$. So in general, you are trying to get a non-function to agree with a non-existent function.