The Theory Of Variable Exponent Function Space And Its Applications

Last decade, more and more researchers have been interested in the theory of the variable exponent function space and its applications. In the beginning, Kovacik and Rakosnik introduced variable exponent Lebesgue and Sobolev spaces as a new method for dealing with nonlinear Dirichet boundary value problems with nonstandard growth and coercive assumption. The variable exponent spaces have an important physics background, and they are used in the study of electrorheological fluids. The study of these spaces has been stimulated by problems of elasticity, fluid dynamics, calculus of variations, and differential equations with p ( x )-growth conditions. Nowadays, variational problems and differential equations with variable exponent are intensively developed by many researchers worldwidely. A lot of results corresponding the researches have been attained. The paper introduces the recent results about the variable exponent function spaces, raises many open problems, and gives some applications on the partial differential equations.In this paper, chapter 1 introduces the background of the topic, and gives many properties and conclusions of the variable exponent function spaces as the preliminary knowledge. Chapter 2 does research on the boundedness of the Hardy-Littlewood maximal operators in the variable exponent Lebesgue spaces, its dual space, and the interpolation problems. Chapter 3 studies the problems of the compactness and the embedding in the variable exponent Sobolev spaces. Chapter 4 gives the conditions about the existence of the solutions of several differential equations in the variable exponent function space.