Abstract
In this paper, we establish the weak lower semicontinuity of variational functionals with variable growth in variable exponent Sobolev spaces. The weak lower semicontinuity is interesting by itself and can be applied to obtain the existence of an equilibrium solution in nonlinear elasticity. 2000 Mathematics Subject Classification: 49A45
Highlights
1 Introduction The main purpose of this paper is to study the weak lower semicontinuity of the functional
If m = n = 1, Tonelli [1] proved that the functional F is lower semicontinuity in W1,∞ (a, b) if and only if the function f is convex in the last variable
This paper is organized as the following: In Section 2, we present some preliminary facts; in Section 3, we discuss the weak lower semicontinuity of variational functionals with variable growth; in Section 4, we give an example to show that the result obtained in Section 3 can be applied to study the existence of an equilibrium solution in nonlinear elasticity
Summary
F(u) = f (x, u, ∇u)dx where Ω is a bounded C1 domain in Rn and f : Rn × Rm × Rnm ® R is a Caratheodory function satisfying variable growth conditions. This paper is organized as the following: In Section 2, we present some preliminary facts; in Section 3, we discuss the weak lower semicontinuity of variational functionals with variable growth; in Section 4, we give an example to show that the result obtained in Section 3 can be applied to study the existence of an equilibrium solution in nonlinear elasticity.
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