Local clustering of the non-zero set of functions in $W^{1,1}(E)$

Abstract

We extend to the $p=1$ case a measure theoretic result previously proved by DiBenedetto and Vespri for functions that belong to $u\in W^{1,p}(K_{\rho }(x_0))$ where $K_{\rho }(x_0))$ is a $N$-dimensional cube of edge $\rho$ centered at $x_0$. It basically states that if the set where $u$ is bounded away from zero occupies a sizable portion of $K_{\rho }$, then the set where $u$ is positive clusters about at least one point of $K_{\rho }$.