Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions
 M. Hintermüller,
 C.Y. Kao,
 A. Laurain
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This paper focuses on the study of a linear eigenvalue problem with indefinite weight and Robin type boundary conditions. We investigate the minimization of the positive principal eigenvalue under the constraint that the absolute value of the weight is bounded and the total weight is a fixed negative constant. Biologically, this minimization problem is motivated by the question of determining the optimal spatial arrangement of favorable and unfavorable regions for a species to survive. For rectangular domains with Neumann boundary condition, it is known that there exists a threshold value such that if the total weight is below this threshold value then the optimal favorable region is like a section of a disk at one of the four corners; otherwise, the optimal favorable region is a strip attached to the shorter side of the rectangle. Here, we investigate the same problem with mixed RobinNeumann type boundary conditions and study how this boundary condition affects the optimal spatial arrangement.
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 Title
 Principal Eigenvalue Minimization for an Elliptic Problem with Indefinite Weight and Robin Boundary Conditions
 Journal

Applied Mathematics & Optimization
Volume 65, Issue 1 , pp 111146
 Cover Date
 20120201
 DOI
 10.1007/s002450119153x
 Print ISSN
 00954616
 Online ISSN
 14320606
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 Asymptotic analysis
 Principal eigenvalue
 Elliptic boundary value problem with indefinite weight
 Robin conditions
 Shape optimization
 Authors

 M. Hintermüller ^{(1)} ^{(2)}
 C.Y. Kao ^{(3)} ^{(4)}
 A. Laurain ^{(5)}
 Author Affiliations

 1. HumboldtUniversity of Berlin, Unter den Linden 6, 10099, Berlin, Germany
 2. Institute of Mathematics and Scientific Computing, University of Graz, Heinrichstrasse 36, 8010, Graz, Austria
 3. Department of Mathematics and Computer Science, Claremont McKenna College, Claremont, CA, 91711, USA
 4. Department of Mathematics, The Ohio State University, 410 Math Tower, 231 West 18th Avenue, Columbus, OH, 432101174, USA
 5. Department of Mathematics, JohannvonNeumannHaus, Rudower Chaussee 25, 12489, BerlinAdlershof, Germany