Pure Mathematics
Vol. 09  No. 06 ( 2019 ), Article ID: 31737 , 7 pages
10.12677/PM.2019.96099

A Rearrangement Optimization Problem Involving a Nonlocal Operator

Chong Qiu1, Yuying Zhou2

1School of Mathematics and Physics, TaiZhou University, Taizhou Jiangsu

2School of Mathematical Sciences, Soochow University, Suzhou Jiangsu

Received: Jul. 21st, 2019; accepted: Jul. 31st, 2019; published: Aug. 16th, 2019

ABSTRACT

In this paper, we study a rearrangement optimization problem involving a nonlocal operator, i.e., fractional Laplacian. Firstly, we use the property of the first eigenvalue to prove that the nonlinear equation with a perturbation term involving the fractional Laplacian has a unique solution. Then, we introduce an optimization problem which takes the ground state energy functional as the objective function. We show that under suitable assumptions such an optimization problem is solvable.

Keywords:Nonlocal Operator, Rearrangement Optimization, Perturbation Term

1泰州学院数理学院，江苏 泰州

2苏州大学数学科学学院，江苏 苏州

1. 引言

$\Omega \subset {R}^{N}$ 为一个有界光滑区域，所谓 $\Omega$ 上可测函数f生成的重排函数空间 $R\left(f\right)$ 是指由所有满足条件：

$meas\left(\left\{x\in \Omega :g\left(x\right)\ge a\right\}\right)=meas\left(\left\{x\in \Omega :f\left(x\right)\ge a\right\}\right),\forall a\in R$

$\left\{\begin{array}{l}-{L}_{\theta }^{s}u=f\left(x\right)+h\left(x,u\right),x\in \Omega \\ u=0,x\in {R}^{N}\\Omega \end{array}$ (1.1)

${L}_{\theta }^{s}u\left(x\right)={\int }_{{R}^{N}}\frac{u\left(x+y\right)+u\left(x-y\right)-2u\left(x\right)}{{|y|}^{N+2s}}\theta \left(y\right)\text{d}y,x\in {R}^{N}$ ,

$\left({P}_{\lambda ,h,f}\right)$ $\left\{\begin{array}{l}{\left(-\Delta \right)}^{s}u=f\left(x\right)+\lambda h\left(x\right)u,x\in \Omega \\ u=0,x\in {R}^{N}\\Omega \end{array}$

$I:{H}^{s}\left(\Omega \right)\to R$ 为方程 $\left({P}_{\lambda ,h,f}\right)$ 的能量泛函，即

$I\left(u\right)=\frac{1}{2}{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{\left(u\left(x\right)-u\left(y\right)\right)}^{2}}{{|x-y|}^{N+2s}}\text{d}x\text{d}y-\frac{\lambda }{2}{\int }_{\Omega }h{u}^{2}\text{d}x-{\int }_{\Omega }fu\text{d}x$ (1.2)

(Opt) $I\left({u}_{\stackrel{^}{h},\stackrel{^}{f}}\right)={\mathrm{inf}}_{h\in R\left({h}_{0}\right),f\in R\left({f}_{0}\right)}I\left({u}_{h,f}\right)$ ?

2. 预备知识

${H}^{s}\left(\Omega \right)=\left\{u\in {L}^{2}\left({R}^{N}\right):u\equiv 0,x\in {R}^{N}\\Omega ,{\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{\left(u\left(x\right)-u\left(y\right)\right)}^{2}}{{|x-y|}^{N+2s}}\text{d}x\text{d}y<\infty \right\}$ ,

$〈u,v〉={\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{|x-y|}^{N+2s}}\text{d}x\text{d}y,\forall u,v\in {H}^{s}\left(\Omega \right)$ .

$u\in {H}^{s}\left(\Omega \right)$ ，则u的范数为

$‖u‖={\left({\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{{\left(u\left(x\right)-u\left(y\right)\right)}^{2}}{{|x-y|}^{N+2s}}\text{d}x\text{d}y\right)}^{\frac{1}{2}}$ .

${\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{|x-y|}^{N+2s}}\text{d}x\text{d}y-{\int }_{\Omega }\left(\lambda huv+fv\right)\text{d}x=0,\forall v\in {H}^{s}\left(\Omega \right)$ .

${I}^{\prime }\left(u\right)v={\int }_{{R}^{N}}{\int }_{{R}^{N}}\frac{\left(u\left(x\right)-u\left(y\right)\right)\left(v\left(x\right)-v\left(y\right)\right)}{{|x-y|}^{N+2s}}\text{d}x\text{d}y-{\int }_{\Omega }\left(\lambda huv+fv\right)\text{d}x,\forall v\in {H}^{s}\left(\Omega \right)$ .

(2.1)

(2.2)

. (2.3)

3. 方程解的存在唯一性

,

, 若,

.

. (2.4)

,

.

.

.

4. 重排优化问题(Opt)的可解性

,

.

. (4.1)

,

.

. (4.2)

. (4.3)

(4.4)

.

.

. (4.5)

. (4.6)

A Rearrangement Optimization Problem Involving a Nonlocal Operator[J]. 理论数学, 2019, 09(06): 755-761. https://doi.org/10.12677/PM.2019.96099

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