NDLI: Properties of optimizers of the principal eigenvalue with indefinite weight and Robin conditions

Content Provider	Springer Nature Link
Author	Nadin, Grégoire Lamboley, Jimmy Laurain, Antoine Privat, Yannick
Copyright Year	2016
Abstract	In this paper, we are interested in the analysis of a well-known free boundary/shape optimization problem motivated by some issues arising in population dynamics. The question is to determine optimal spatial arrangements of favorable and unfavorable regions for a species to survive. The mathematical formulation of the model leads to an indefinite weight linear eigenvalue problem in a fixed box $$\Omega $$ and we consider the general case of Robin boundary conditions on $$\partial \Omega $$ . It is well known that it suffices to consider bang-bang weights taking two values of different signs, that can be parametrized by the characteristic function of the subset E of $$\Omega $$ on which resources are located. Therefore, the optimal spatial arrangement is obtained by minimizing the positive principal eigenvalue with respect to E, under a volume constraint. By using symmetrization techniques, as well as necessary optimality conditions, we prove new qualitative results on the solutions. Namely, we completely solve the problem in dimension 1, we prove the counter-intuitive result that the ball is almost never a solution in dimension 2 or higher, despite what suggest the numerical simulations. We also introduce a new rearrangement in the ball allowing to get a better candidate than the ball for optimality when Neumann boundary conditions are imposed. We also provide numerical illustrations of our results and of the optimal configurations.
Ending Page	37
Page Count	37
Starting Page	1
File Format	PDF
ISSN	09442669
e-ISSN	14320835
Journal	Calculus of Variations and Partial Differential Equations
Issue Number	6
Volume Number	55
Language	English
Publisher	Springer Berlin Heidelberg
Publisher Date	2016-11-03
Publisher Place	Berlin, Heidelberg
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Methods based on necessary conditions Systems Theory, Control Calculus of Variations and Optimal Control; Optimization Analysis Theoretical, Mathematical and Computational Physics Optimal control problems involving ordinary differential equations Variational methods for eigenvalues of operators Problems involving partial differential equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis

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