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Rendiconti Lincei - Matematica e Applicazioni

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Volume 17, Issue 3, 2006, pp. 223–225
DOI: 10.4171/RLM/465

Published online: 2006-09-30

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

Emmanuele DiBenedetto[1], Ugo Gianazza[2] and Vincenzo Vespri[3]

(1) Vanderbilt University, Nashville, United States
(2) Università di Pavia, Italy
(3) Universita di Firenze, Italy

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 $K_\rho$.

Keywords: $W^{1,1}$ functions; measure theory; positivity set

DiBenedetto Emmanuele, Gianazza Ugo, Vespri Vincenzo: Local clustering of the non-zero set of functions in $W^{1,1}(E)$. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 17 (2006), 223-225. doi: 10.4171/RLM/465