Relative Rearrangement And Its Application In Partial Differential Equations

This thesis deals with the Relative rearrangement and its applications in partial. differential equations.The thesis composes of five chapters:In Chapter 1 we summarize the background of the related issues and state the main results of the present thesis. We also give some preliminary results and notations used in the whole thesis.Chapter 2 consider an nonlocal problem arising in the confinement of a plasma in a current carrying stellarator. By using the properties of relative rearrangement, we start by reformulating (P) into an equivalent problem ((?)), eliminating the unknown function F. Then we prove some convergence results about the nonlocal term u*′(│u> u(x)│) and u+*′(│u> u(x)│), and obtain the existence of solutions to problem ((?)) by the Galerkin method.Chapter 3 mainly discuss a nonlocal model arising in plasma physics and obtain the existence of solutions by using the Galerkin method and topological degree theory.Chapter 4 studies a nonlocal model for a confined plasma in a Tokamak and prove the existence of solutions by twice approximation and the topological degree theory. We also obtain the existence of free boundary to such problem. In order to prove these results, we have established some properties about the relative rearrangement and the derivative of the decreasing rearrangement.Chapter 5 deals with a type of degenerate quasi-linear elliptic equations. In order to get the existence of solutions, we obtain-a prior up bounds for the L∞norm of solutions by using the theory of relative rearrangement. Moreover, we introduce the notion of renormalized solutions and prove the existence of renormalized solutions.