THE DIRICHLET ENERGY INTEGRAL AND VARIABLE EXPONENT SOBOLEV SPACES WITH ZERO BOUNDARY VALUES P. HARJULEHTO, P. HÄSTÖ, M. KOSKENOJA, ANDS. VARONEN A . We define and study variable exponent Sobolev spaces with zero bound- ary values. This allows us to prove that the Dirichlet energy integral has a minimizer in the variable exponent case. Our results are based on a Poincaré-type inequality, which we prove under a certain local jump condition for the variable exponent. 1. I In the beginning of the 90’s Kovᡠcik and Rákosník [KR] introduced variable expo- nent Lebesgue and Sobolev spaces. In fact, these spaces are special cases of so-called Orlicz-Musielak spaces, and in this form their investigation goes back a bit further, to Hudzik [Hud] and Musielak [Mus]. During the last decade Sobolev spaces with vari- able exponent have been studied intensively by Diening [Die], Diening and R˚ užiˇ cka [DR], Edmunds and Rákosník [ER1, ER2, ER3], Fan, Shen, and Zhao [FSZ], and Pick and R ˚ užiˇ cka [PR], among others. One area where these spaces have found applications is the study of electrorheo- logical fluids, as described in the book of R˚ užiˇ cka [Ruz]. The same spaces appear also in the study of variational integrals with non-standard growth, see the papers by Zhikov [Zhi], Maracellini [Mar], and Acerbi and Mingione [AM]. The classical Dirichlet boundary value problem arises from a partial differential equation; if Ω is a domain in  n and w : ∂Ω →  is a continuous function, then the problem is to find a continuous function u : Ω →  so that the Laplace equation −Δu = 0 is satisfied on Ω and u = w on ∂Ω. The function w gives the boundary values of u. By Weyl’s lemma, such a u is always a C 2 -function on Ω, and hence the problem may be considered in the classical sense. Classical potential theory is based on the Laplace equation which is clearly linear. The p-Dirichlet boundary value problem for fixed p,1 < p < ∞, is to find a continuous function u on Ω so that the p-Laplace equation (1.1) − div(|∇u| p−2 ∇u) = 0 is satisfied on Ω and u = w on ∂Ω. Even more generally, we search for a function u ∈ W 1, p (Ω) and the boundary values are given with w ∈ W 1, p (Ω) only in the Sobolev sense, that is, u − w ∈ W 1, p 0 (Ω). The p-Laplace equation (1.1) is the Euler equation for the variational integral (1.2)  Ω |∇u| p dx Date: 16.4.2003. 2000 Mathematics Subject Classification. 46E35, 31C45, 35J65. Key words and phrases. Variable exponent Sobolev space, zero boundary values, Sobolev capac- ity, Poincaré inequality, Dirichlet energy integral, minimizing problem. 1