Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains
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Department of Mathematics, The Ohio State University, Columbus, OH 43210
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Department of Mathematics, The Ohio State State University, Columbus, Ohio 43210
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Mathematical Institute, Tohoku University, Sendai 980-8578
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Received:
01 November 2007
Accepted:
29 June 2018
Published:
01 March 2008
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MSC :
35P15, 35J20, 92D25.
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This paper is concerned with an indefinite weight linear eigenvalue
problem in cylindrical domains. We investigate the minimization of the positive
principal eigenvalue under the constraint that the weight is bounded by
a positive and a negative constant 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. Both our analysis and numerical simulations
for rectangular domains indicate that there exists a threshold value such that
if the total weight is below this threshold value, then the optimal favorable
region is a circular-type domain at one of the four corners, and a strip at the
one end with shorter edge otherwise.
Citation: Chiu-Yen Kao, Yuan Lou, Eiji Yanagida. Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains[J]. Mathematical Biosciences and Engineering, 2008, 5(2): 315-335. doi: 10.3934/mbe.2008.5.315
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Abstract
This paper is concerned with an indefinite weight linear eigenvalue
problem in cylindrical domains. We investigate the minimization of the positive
principal eigenvalue under the constraint that the weight is bounded by
a positive and a negative constant 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. Both our analysis and numerical simulations
for rectangular domains indicate that there exists a threshold value such that
if the total weight is below this threshold value, then the optimal favorable
region is a circular-type domain at one of the four corners, and a strip at the
one end with shorter edge otherwise.
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