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ArticleOriginal scientific text

Title

A limit involving functions in $W^{1, p}_0 (Ω)$

Authors ¹

Affiliations

Department of Mathematicsi, University of Catania, Viale A. Doria 6, 95125 Catania, Italy

Abstract

We point out the following fact: if Ω ⊂

ℝ^{n}

is a bounded open set, δ>0, and p>1, then

lim_{\to 0^{+}} \in f_{\in V_{□}} \int_{Ω} {| \nabla (x) |}^{p} d x = \infty

, where

V_{=} {\in W^{1, p}_0 (Ω) : m e a s ({x \in Ω | (x) ∣ > δ}) >} .

H. Brézis, Analyse fonctionnelle, Masson, 1983.
V. G. Maz'ja, Sobolev Spaces, Springer, 1985.