首页 | 本学科首页   官方微博 | 高级检索  
     检索      


Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form
Institution:1. Politecnico di Torino, Dipartimento di Scienze Matematiche “G.L. Lagrange”, Corso Duca degli Abruzzi, 24, 10129, Torino, Italy;2. CNRS, Sorbonne Université, Inria, Université de Paris, Laboratoire Jacques-Louis Lions, Paris, France;3. Dipartimento di Matematica Tullio Levi-Civita, Università degli Studi di Padova, via Trieste 63, 35131 Padova, Italy;4. Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes, 91190, Gif-sur-Yvette, France;1. Department of Mathematics, Texas A & M University, College Station, TX 77843-3368, USA;2. Department of Mathematics, University of North Carolina - Chapel Hill, Chapel Hill, NC 27599, USA;3. Department of Mathematics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada;1. Yau Mathematical Sciences Center, Tsinghua University, Beijing 100084, China;2. Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China;1. EPFL SB, Station 8, CH-1015 Lausanne, Switzerland;2. Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland;3. IMATI-CNR, via Ferrata 5, I-27100 Pavia, Italy;1. Instituto de Matemática, Universidade Federal do Rio de Janeiro, C. P. 68.530, CEP 21.945-970, Rio de Janeiro, RJ, Brazil;2. Department of Mathematics, University of Oklahoma, Norman, OK, USA;3. Department of Mathematics, Southern University of Science and Technology of China, Guangdong, China;4. Departamento de Geometria, Instituto de Matemática e Estatística, Universidade Federal Fluminense, Niterói, Brazil
Abstract:In this paper we consider minimizers of the functionalmin?{λ1(Ω)+?+λk(Ω)+Λ|Ω|,:Ω?D open} where D?Rd is a bounded open set and where 0<λ1(Ω)?λk(Ω) are the first k eigenvalues on Ω of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets Ω? have finite perimeter and that their free boundary ?Ω?D is composed of a regular part, which is locally the graph of a C1,α-regular function, and a singular part, which is empty if d<d?, discrete if d=d? and of Hausdorff dimension at most d?d? if d>d?, for some d?{5,6,7}.
Keywords:Shape optimization  Eigenvalues  Operator in divergence form  Optimality conditions  Regularity of the free boundaries  Viscosity solutions
本文献已被 ScienceDirect 等数据库收录!
设为首页 | 免责声明 | 关于勤云 | 加入收藏

Copyright©北京勤云科技发展有限公司  京ICP备09084417号