Application Of Gauss Symmetrization In Elliptic And Parabolic Equations

This thesis deals with elliptic and parabolic equations related to Gauss measure. The topic includes the following aspects:to establish conditions for the existence of Gauss symmetric solutions to the "symmetrized" equations which define on half spaces and whose data depend only on the first variable; to compare the solutions of the original equations with the ones of the "symmetrized" equations and concern the necessary and sufficient conditions for equalities of these comparisons; to utilize the comparison results to study the regularity and existence of the solutions.Chapter 1 is to summarize the background of the related issues and state the main results of the present thesis.Chapter 2 is to introduce some theories of Gauss symmetrization. n.Chapter 3 considers a degenerate elliptic equations -ΣDj(aijDiu)+cu= gφ-i, j=1 div(fφ), whereφis the Gauss density. By comparing with the solutions to the "sym-metrized" equations, we explore how the data influence the regularity of the solutions.Chapter 4 is devoted to study elliptic equations Au= fφand parabolic equations ut+Firstly,we explorethe existence of Gauss symmetric solutions to several "symmetrized" equations. Secondly, under different assumptions, we study the comparisons between the original equations and the corresponding "symmetrized" equations, and then investigate the necessary and sufficient conditions for equalities of these comparisons.Chapter 5 deals with a class of Dirichlet problems for nonlinear elliptic equations-div(a(x,u, Du))= H(x,u,Du). A condition is exhibited to ensure the existence of Gauss symmetric solutions to the "symmetrized" equations. Moreover, we consider the comparisons between the solutions to the original equations and the "symmetrized" equa-tions and study the necessary and sufficient conditions for equalities of these comparisons. Finally, we obtain the existence of the solutions.