Local clustering of the non-zero set of functions in \(W^{1,1} (E)\)
Summary: We extend to the \(p=1\) case a measure theoretic result previously proved by \textit{E. DiBenedetto} and \textit{V. Vespri} [Arch. Ration. Mech. Anal. 132, No. 3, 247--309 (1995; Zbl 0849.35060)] 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 seizable portion of \(K_{\rho}\), then the set where \(u\) is positive clusters about at least one point of \(K_{\rho}\).
scientific article; zbMATH DE number 5177497