Applications Of Steiner Symmetrization To Elliptic And Parabolic Equations

It is well-known that rearrangement methods have turned out to be a useful tool for the study of elliptic and parabolic equations. Rearrangement methods are also called the sym-metrization methods. The symmetrization methods were systematically introduced by Talenti. There are many extensions and generalization of his results. Most of their results are based on Schwarz symmetrization. This thesis deals with partial differential equations by using Steiner symmetrization. The Schwarz symmetrization is a symmetrization with respect to a point. So it maybe loses information about the properties that arise from the symmetry of the data with respect to axes or planes. We will give the symmetrization problem of the origin problem and establish the relationship of the solution between the symmetrization problem and the origin problem by using Steiner symmetrization.The thesis is composed of five chapters:In chapter 1, we summarize the backgrounds of rearrangement methods of the partial dif-ferential equations and state the main results of the thesis. We also give out some preliminary results used in the whole thesis.In chapter 2, we consider the elliptic Neumann problem where the coefficient of the zero order term contains x by using Steiner symmetrization. We get the symmetrization Dirichlet-Neumann problem and establish the comparison result between the solutions.In chapter 3, we consider the sublinear elliptic problem by using Steiner symmetrization. Firstly, we get the L~? estimate of the origin problem. Then we obtain that the symmetrization problem of the origin problem is a linear problem. Finally we establish the comparison result between the solutions and give the energy estimate inequality.In chapter 4, we consider the sublinear parabolic problem by using Steiner symmetrization. Firstly, we get the L~? estimate of the origin problem. Then we obtain the symmetrization problem of the origin problem and establish the comparison result between the solutions. Finally we give the energy estimate inequality.In chapter 5, we give the conclusion and prospect of this thesis.