Abstract
This paper is concerned with the functionalJdefined byJ(u)=∫Ω×ΩW(x,y,∇u(x),∇u(y))dx dy, whereΩ⊂ℝNis a regular open bounded set andWis a real-valued function with variable growth. After discussing the theory of Young measures in variable exponent Sobolev spaces, we study the weak lower semicontinuity and relaxation ofJ.
Highlights
In this paper, we are concerned with a type of variational integrals that can be written asJ (u) = ∫ W (x, y, ∇u (x), ∇u (y)) dx dy, (1)Ω×Ω where Ω ⊂ RN is a regular open bounded set (N ≥ 1), u ∈ W1,p(x)(Ω; Rm), and p(x) is Lipschitz continuous with 1 < p− := infx∈Ωp(x) ≤ p(x) ≤ p+ := supx∈Ωp(x) < ∞
After Kovacik and Rakosnık first discussed Lp(x)(Ω) and Wm,p(x)(Ω) spaces in [1], a lot of research has been done concerning these kinds of variable exponent spaces; for examples, see [2,3,4,5,6] for the properties of such spaces and [7,8,9] for the applications of variable exponent spaces on partial differential equations
The main goal of this paper is to study the weak lower semicontinuity and relaxation of the nonlocal variational problems
Summary
We are concerned with a type of variational integrals that can be written as. After Kovacik and Rakosnık first discussed Lp(x)(Ω) and Wm,p(x)(Ω) spaces in [1], a lot of research has been done concerning these kinds of variable exponent spaces; for examples, see [2,3,4,5,6] for the properties of such spaces and [7,8,9] for the applications of variable exponent spaces on partial differential equations These problems with variable exponent growth possess very complicated nonlinearities; for instance, the p(x)-Laplacian operator is inhomogeneous. The main goal of this paper is to study the weak lower semicontinuity and relaxation of the nonlocal variational problems We will analyze these problems in terms of Young measures generated by sequences in variable exponent space.
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