In (6) (see also 7), Sobolev introduced a class of function spaces Wm,p(Ω) (m a non-negative integer, 1 < p < ∞) defined on open subsets Ω of Euclidean space En, which have important applications in partial differential equations. They are defined as follows. For each n-tuple α = (α1, … αn) of non-negative integers let
The spaces with a random variable exponent and are introduced. After discussing the properties of the spaces and , we give an application of these spaces to the stochastic partial differential equations with random variable growth.
Function Spaces with a Random Variable Exponent