Study On Some Rearrangement Optimization Problems Related To Nonlinear Elliptic Equations

In this thesis we discuss the rearrangement optimization problems as the following form: (P1):min{Ψg):g∈R(f)} or (P2):max{Ψ(g):g∈R(f)}, where f is a measurable function defined in the bounded domain Ω (?) RN,R(f) is the set of all the rearrangement functions of f,Ψ:R(f)(?)R is some cost functional respectively as the energy functionals of the following equations and the first eigenvalue corresponding to the following equation: where Δpu= div{|▽u|p-2Δu) is the p-Laplacian, divA(x,▽u) is the general divergence-type operator, Lθs is a non-local operator and (-Δ)s is the fractional Laplace operator.In Chapters 2 and 3, we prove under different conditions that both the rearrange-ment optimization problems (P1) and (P2) corresponding to the energy functionals of Equations (Ⅰ) and (Ⅱ) are solvable. Moreover, if Ω= B(0,r) then the solution of the problem (P1) is symmetric.In Chapter 4, by using the results of the variational inequality we prove that the problem (P1) corresponding to the energy functional of Equation (Ⅲ) is solvable.In Chapter 5, we use some methods from dealing with the non-local operator to obtain the solvability of both the problems (P1) and (P2) corresponding to the energy functional of Equation (Ⅳ).In Chapter 6, we prove that both the rearrangement optimization problems (P1) and (P2) corresponding to the first eigenvalue of Equation (V) are solvable. Moreover, if Ω=B(0, r) then the solution of the problem (P1) is symmetric.