| Recently, a great number of nonlinear problems with variable growth, such as the electrorheological fluid motion model, arises in natural science and engineering. Con-sequently, the classical Lebesgue and Sobolev spaces show their limitations in the study of partial differential equations. So we need to study variable exponent Lebesgue and Sobolev spaces, i.e. Lp(x) and W1,p(x) spaces. Currently, researchers in growing number show great concern about studying the problems with variable growth.After Young first proposed Young measures, Young measures are becoming a pow-erful tool in treating weak convergence and non-convex variational problems. In addition, Young measures have important applications in the study of partial differential equations, continuum mechanics and ferromagnetism.In this dissertation, on the basis of the theory of variable exponent spaces, we will study the basic theorems and properties of Young measures generated by sequences in variable exponent Lebesgue and Sobolev spaces. Then we use this tool to study some problems with variable growth.The main work of this dissertation is as follows:Firstly, the basic theorems of Young measures generated by sequences in variable exponent Lebesgue and Sobolev spaces will be considered. We will prove that the bound-ed sequences in variable exponent Lebesgue and Sobolev spaces can generate a family of Young measures. Particularly, the bounded sequences in variable exponent Sobolev space and its gradients can generate a product measure of a Dirac measure and a W1,p(x)-Young measure.Secondly, we will discuss the basic properties of Young measures generated by se-quences in variable exponent spaces. On this basis, we will study the weak lower semi-continuity and relaxation of a nonlocal variational problem. Two sufficient and necessary conditions of weak lower semicontinuity will be given and the minimum and minimizer of the relaxation will be studied.Then we will study the existence of solutions for quasilinear elliptic systems in di-vergence form with variable growth under four kinds of monotonicity condition. We will show an example which is monotone but not strictly monotone so that the method of monotone operators is not valid. A Galerkin approximation sequence will be estab-lished by the method of Galerkin approximation. We will use the Young measures gen-erated by the Galerkin approximation sequence to prove that the weak limit of approx-imation sequence is the weak solution of the equations. At last, an example of strictly p(x)-quasimonotone function will be given.Finally, we will study the existence of solutions for quasilinear parabolic systems in divergence form with variable growth under four kinds of monotonicity condition. We will choose an L2-orthonormal base and make a Galerkin approximation sequence. Then we will use the Young measures generated by the Galerkin approximation sequence to prove that the weak limit of approximation sequence is the weak solution of the equation. |
