Converse Symmetry and Intermediate Energy Values in Rearrangement Optimization Problems

Yichen Liu,Behrouz Emamizadeh

doi:10.1137/16m1100307

Abstract

This paper discusses three rearrangement optimization problems where the energy functional is connected with the Dirichlet or Robin boundary value problems. First, we consider a simple model of Dirichlet type, derive a symmetry result, and prove an intermediate energy theorem. For this model, we show that if the optimal domain (or its complement) is a ball centered at the origin, then the original domain must be a ball. As for the intermediate energy theorem, we show that if $\alpha,\beta$ denote the optimal values of corresponding minimization and maximization problems, respectively, then every $\gamma$ in $(\alpha,\beta)$ is achieved by solving a max-min problem. Second, we investigate a similar symmetry problem for the Dirichlet problems where the energy functional is nonlinear. Finally, we show the existence and uniqueness of rearrangement minimization problems associated with the Robin problems. In addition, we shall obtain a symmetry and a related asymptotic result.

Highlights

A rearrangement optimization problem is an optimization problem of the following forms: (1.1)inf Φ(f ) and sup Φ(f ), f ∈R(f0) f ∈R(f0)where R(f0) is a rearrangement class1 generated by f0, a prescribed function
Where R(f0) is a rearrangement class1 generated by f0, a prescribed function
The goal function Φ is frequently a nonlinear functional which arises from a boundary value problem

Summary

Introduction

A rearrangement optimization problem is an optimization problem of the following forms:. CONVERSE SYMMETRY AND INTERMEDIATE ENERGY VALUES physical setting, the function uf denotes the displacement, whence, the optimization problems (1.1) address the question of finding the level of vulnerability of the membrane relative to rearranged force in R(f0). Motivated by the abovementioned papers, other authors (see, for example, [10, 15, 25]), studied the same problems (1.1) except that uf is the solution of the following boundary value problem. The last part of the paper is devoted to a rearrangement minimization problem associated with the following Robin boundary value problem:.

Since f

Passing to limit in the above equation yields