Eigenvalue problems in anisotropic spaces

CHAPTER 1 General introduction The motivation for the whole work is nding a solution of the following dierential problem: (P ) ( div (jruj p(x) 2 jruj) = juj p(x) 2 u in u = 0 on @ Partial dierential equations and variational problems with p(x)-growth conditions arise from physics, in particular from problems involving electrorheological uids, thermorheological uids, image processing and nonlinear elasticity theory. The usual approach to a dierential problem of this kind, following a basic tool of analysis, is that of considering its variational formulation. So the natural framework for u turns to be a certain variable exponent Sobolev space W 1;p(x) . In Chapter 2 we recall basic concepts concerning variable exponent spaces. Chapters 3 and 4 are devoted to characterize the problem, showing some existence results and some interesting properties that (u; ) must have. Problem (P ) is an extreme generaliza- tion of the Helmholtz equation which was proposed in the mid nineteenth century. In these chapters we will see that (P ) can be joint with a inf-problem of Rayleigh quotient. In Chapter 5 we will propose an algorithm to computeu and with a certain approximation error. The main structure of the algorithm is based on the classical inverse power method. The diculties in this task are multiple but, once they are exceeded, we can appreciate the results. Chapter 6 is divided in two parts: the rst one compares the results we have obtained with constant exponent to some results of a brilliant article [9]; the second part shows a few innovative outcomes that we have reached with variable exponent, that has no precedented in literature. 1