Principal eigenvalue for an elliptic problem with indefinite weight on cylindrical domains - Chiu-Yen Kao; Yuan Lou; Eiji Yanagida (Mathematical Biosciences and Engineering)
This paper is concerned with an indefinite weight linear eigenvalueproblem in cylindrical domains. We investigate the minimization of the positiveprincipal eigenvalue under the constraint that the weight is bounded bya positive and a negative constant and the total weight is a fixed negativeconstant. Biologically, this minimization problem is motivated by the questionof determining the optimal spatial arrangement of favorable and unfavorableregions for a species to survive. Both our analysis and numerical simulationsfor rectangular domains indicate that there exists a threshold value such thatif the total weight is below this threshold value, then the optimal favorableregion is a circular-type domain at one of the four corners, and a strip at theone end with shorter edge otherwise.