Variable Sobolev capacity and the assumptions on the exponent

Abstract

In the beginning of the 90’s Kovacik and Rakosnik [17] introduced variable exponent Lebesgue and Sobolev spaces. In fact, generalized Lebesgue and Sobolev spaces are special cases of so-called Orlicz-Musielak spaces, and in this form their investigation goes back a bit further, to Orlicz [20], Hudzik [15], and Musielak [18], see also Sharapudinov [23]. During the last couple of years Lebesgue and Sobolev spaces with variable exponent have been studied at an increasing pace by Diening [4, 5], Edmunds and Rakosnik [6, 7], Fan, Shen and Zhao [9, 10], Cruz-Uribe, Fiorenze and Neugebauer [3], Kokilasvili and Samko [16], and Nekvinda [19], among others. One area where these spaces have found applications is the study of electrorheological fluids, as described in the book of Růžicka [22]. A mathematical application is the study of variational integrals with non-standard growth, see the papers by Acerbi and Mingione [1, 2]. Sobolev capacity for fixed exponent spaces has found a great number of uses (e.g. the monographs by Evans and Gariepy [8] and Heinonen, Kilpelainen, and Martio [14]). It was introduced into the study of variable exponent spaces in [12] and has been applied to the investigation of zero boundary values of Sobolev functions in [13]. In [12] we required the assumption 1 < ess inf p ≤ ess sup p < ∞ of the variable exponent p to guarantee that our set-function is indeed a Choquet capacity. This is unsatisfactory, since there is no reason to expect this condition to be of relevance in this context. In this paper we show that the lower inequality needs to hold only locally. In particular we show in Corollary 4.2 that if the exponent p is continuous, then zero capacity sets enjoy the usual subadditivity property.