1,720,990 research outputs found
Maximization of the first eigenvalue in problems involving the bi-Laplacian
No full textThis paper concerns maximization of the first eigenvalue in problems involving the bi-Laplacian under either Navier boundary conditions or Dirichlet boundary conditions. Physically, in the case of N = 2, our equation models the vibration of a nonhomogeneous 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 throughout Ω so as to maximize the first eigenvalue in the vibration of the corresponding plate
Steiner symmetry in the minimization of the first eigenvalue in problems involving the p-Laplacian
No full textLet Ω ⊂ RN be an open bounded connected set. We consider the eigenvalue problem −Δpu = λρ|u|p−2u in Ω with homogeneous Dirichlet boundary condition, where Δp is the p-Laplacian operator and ρ is an arbitrary function that takes only two given values 0 < α < β and that is subject to the constraint ∫Ω ρdx = αγ +β(|Ω|−γ) for a fixed 0 < γ < |Ω|. The optimization of the map ρ ↦ λ1(ρ), where λ1 is the first eigenvalue, has been studied by Cuccu, Emamizadeh and Porru. In this paper we consider a Steiner symmetric domain Ω and we show that the minimizers inherit the same symmetry
Symmetry breaking in the minimization of the second eigenvalue for composite membranes
No full textLet Ω C RN be an open bounded connected set. We consider the eigenvalue problem -Δu = λρu in Ω with Dirichlet boundary condition, where ρ is an arbitrary function that assumes only two given values 0 < α < β and is subject to the constraint ∫ Ω ρ dx = αγ + β(|Ω| - γ) for a fixed 0 < γ < |Ω|. Cox and McLaughlin studied the optimization of the map ρ → λκ(ρ), where λκ is the kth eigenvalue. In this paper we focus our attention on the case when N ≥ 2, k = 2 and Ω is a ball. We show that, under suitable conditions on α, β and γ, the minimizers do not inherit radial symmetry
Global symplectic coordinates on complex domains
No full textIn this paper we deal with complex domains M ⊂ Cn equipped with a Ka ̈hler form ω = i ∂∂ ̄f, 2
where f : M → R only depends on |zj|2, j = 1,...,n for the complex coordinates (z1,...,zn) in Cn. We give an explicit symplectic immersion Φ of (M,ω) into R2n in Section 2. In Section 3 we study when the map Φ is a global symplectomorphism for the case of complete Reinhardt domains in C2
Symmetry breaking in the minimization of the first eigenvalue for the composite clamped punctured disk
No full textLet D0={x∈R2:0<|x|<1} be the unit punctured disk. We consider the first eigenvalue λ1(ρ) of the problem Δ2u=λρu in D0 with Dirichlet boundary condition, where ρ is an arbitrary function that takes only two given values 0<α<β and is subject to the constraint ∫D0ρdx=αγ+β(|D0|−γ) for a fixed 0<γ<|D0|. We will be concerned with the minimization problem ρ↦λ1(ρ). We show that, under suitable conditions on α, β and γ, the minimizer does not inherit the radial symmetry of the domain
Symmetry of solutions to optimization problems related to partial differential equations
No full textWe investigate maxima and minima of some functionals associated with solutions to Dirichlet problems for elliptic equations. We prove
existence results and, under suitable restrictions on the data, we show that any maximal configuration satisfies a special system of two equations. Next, we use the moving plane method to find symmetry results
for solutions of a system. We apply these results to discuss symmetry for the maximal configurations of the previous problem
Balanced metrics on C^n
No full textLet g be a Kaehler metric on C^n.
In this paper we prove that if g is rotation invariant and balanced (in the sense of Donaldson) then, up to biholomorphic isometries,
g equals the Euclidean metric.
The proof of our theorem is based on Calabi's diastasis function and on the characterization of the exponential function due to Miles and Williamson
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