Some Recent Results Regarding Symmetry and Symmetry-breaking Properties of Optimal Composite Membranes

Pedrosa, Renato H. L.

doi:10.1007/3-7643-7401-2_29

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2022

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Cited by 2 publications

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Variação primeira e segunda para o primeiro autovalor de um problema elíptico

Martins¹

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We will consider the elliptic problem −∆u + αχ D u = λu in Ω, where Ω is a domain in R n with regular boundary, and D ⊂ Ω is a closed subset with prescribed Lebesgue measure. The motivation for this problem comes from Mechanics, where this equation models the vibrations of a composite membrane. Let λ 1 (D) be the first eigenvalue of the problem, which is seen as a function of the set D. In this work, we will show that λ 1 is a simple eigenvalue, and we will study the problem of minimizing λ 1 (D) when D varies in the family of all closed subsets of Ω with a given Lebesgue measure. More precisely, we will determine formulas for the first and the second variation of λ 1 .

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Variação primeira e segunda para o primeiro autovalor de um problema elíptico

Martins¹

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Finite element method for an eigenvalue optimization problem of the Schrödinger operator

Guo¹,

Lu²,

Zhang

2022

MATH

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<abstract><p>In this paper, we study the optimization algorithm to compute the smallest eigenvalue of the Schrödinger operator with volume constraint. A finite element discretization of this problem is established. We provide the error estimate for the numerical solution. The optimal solution can be approximated by a fixed point iteration scheme. Then a monotonic decreasing algorithm is presented to solve the eigenvalue optimization problem. Numerical simulations demonstrate the efficiency of the method.</p></abstract>

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Some Recent Results Regarding Symmetry and Symmetry-breaking Properties of Optimal Composite Membranes

Cited by 2 publications

References 10 publications

Variação primeira e segunda para o primeiro autovalor de um problema elíptico

Variação primeira e segunda para o primeiro autovalor de um problema elíptico

Finite element method for an eigenvalue optimization problem of the Schrödinger operator

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