On a Rayleigh-Faber-Krahn Inequality for the Regional Fractional Laplacian

作     者：Tianling Jin Dennis Kriventsov Jingang Xiong 

作者机构：Department of MathematicsThe Hong Kong University of Science and TechnologyClear Water BayKowloonHong Kong Department of MathematicsRutgers University110 Frelinghuysen RoadPiscatawayNJ 08854USA School of Mathematical SciencesLaboratory of Mathematics and Complex SystemsMOEBeijing Normal UniversityBeijing 100875China 

出 版 物：《Annals of Applied Mathematics》 (应用数学年刊（英文版）)

年 卷 期：2021年第37卷第3期

页      面：363-393页

学科分类：07[理学] 0701[理学-数学] 070101[理学-基础数学] 

基　　金：supported by Hong Kong RGC grants ECS 26300716 and GRF 16302519 partially supported by NSFC 11922104 and 11631002 

主　　题：Rayleigh-Faber-Krahn inequality regional fractional Laplacian first eigen・value 

摘      要：We study a Rayleigh-Faber-Krahn inequality for regional fractional Laplacian *** particular,we show that there exists a compactly supported nonnegative Sobolev function u_(0)that attains the infimum(which will be a positive real number)of the set{{∫∫(u0)×(u0)|u(x)-u(y)|^(2)/|x-y|^(n+2σ)dxdy:u∈^(σ)(R^(n)),∫R^(n)u^(2)=1,|{u0}|≤1}.Unlike the corresponding problem for the usual fractional Laplacian,where the domain of the integration is R^(n)×R^(n),symmetrization techniques may not apply *** approach is instead based on the direct method and new a priori diameter *** also present several remaining open questions concerning the regularity and shape of the minimizers,and the form of the Euler-Lagrange equations.