Local boundedness for minimizers of convex integral functionals in metric measure spaces

Abstract

In this paper we consider the convex integral functional I:=∫ΩΦ(gu)dμ in the metric measure space (X,d,μ), where X is a set, d is a metric, µ is a Borel regular measure satisfying the doubling condition, Ω is a bounded open subset of X, u belongs to the Orlicz-Sobolev space N1,Φ(Ω), Φ is an N-function satisfying the Δ2-condition, gu is the minimal Φ-weak upper gradient of u. By improving the corresponding method in the Euclidean space to the metric setting, we establish the local boundedness for minimizers of the convex integral functional under the assumption that (X,d,μ) satisfies the (1,1)-Poincare inequality. The result of this paper can be applied to the Carnot-Caratheodory space spanned by vector fields satisfying Hormander's condition.