This paper concerns minimization and maximization of the first eigenvalue in problems involving the p-Laplacian, under homogeneous Dirichlet boundary conditions. Physically, in the case of N = 2 and p close to 2, our equation models the vibration of a nonhomogeneous membrane Ω which is fixed along the boundary. Given several materials (with different densities) of total extension |Ω|, we investigate the location of these material inside Ω so as to minimize or maximize the first mode in the vibration of the membrane.
This paper concerns the minimization of the first eigenvalue in problems involving the bi-Laplacian under either homogeneous Navier boundary conditions or homogeneous Dirichlet boundary conditions. Physically, in case of N = 2, our equation models the vibration of a non homogeneous plate Ω which is either hinged or clamped along the boundary. Given several materials (with different densities) of total extension |Ω|, we investigate the location of these materials inside Ω so to minimize the first mode in the vibration of the corresponding plate.
We investigate minimization and maximization of the principal eigenvalue of the Laplacian under Dirichlet boundary conditions in case the weight has indefinite sign and varies in a class of rearrangements. Biologically, such optimization problems are motivated by the question of determining the most convenient spatial arrangement of favorable and unfavorable resources for a species to survive or to decline. The question may have practical importance in the context of reserve design or pest control.
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