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Bound states of a converging quantum waveguide

Giuseppe Cardone, Sergei A. Nazarov, Keijo Ruotsalainen (2013)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

We consider a two-dimensional quantum waveguide composed of two semi-strips of width 1 and 1 − ε, where ε > 0 is a small real parameter, i.e. the waveguide is gently converging. The width of the junction zone for the semi-strips is 1 + O(√ε). We will present a sufficient condition for the existence of a weakly coupled bound state below π2, the lower bound of the continuous spectrum. This eigenvalue in the discrete spectrum is unique and its asymptotics is constructed and justified when ε → 0+....

Global minimizer of the ground state for two phase conductors in low contrast regime

Antoine Laurain (2014)

ESAIM: Control, Optimisation and Calculus of Variations

The problem of distributing two conducting materials with a prescribed volume ratio in a ball so as to minimize the first eigenvalue of an elliptic operator with Dirichlet conditions is considered in two and three dimensions. The gap ε between the two conductivities is assumed to be small (low contrast regime). The main result of the paper is to show, using asymptotic expansions with respect to ε and to small geometric perturbations of the optimal shape, that the global minimum of the first eigenvalue...

Minimising convex combinations of low eigenvalues

Mette Iversen, Dario Mazzoleni (2014)

ESAIM: Control, Optimisation and Calculus of Variations

We consider the variational problem         inf{αλ1(Ω) + βλ2(Ω) + (1 − α − β)λ3(Ω) | Ω open in ℝn, |Ω| ≤ 1}, for α, β ∈ [0, 1], α + β ≤ 1, where λk(Ω) is the kth eigenvalue of the Dirichlet Laplacian acting in L2(Ω) and |Ω| is the Lebesgue measure of Ω. We investigate for which values of α, β every minimiser is connected.

Optimal design of turbines with an attached mass

Boris P. Belinskiy, C. Maeve McCarthy, Terry J. Walters (2003)

ESAIM: Control, Optimisation and Calculus of Variations

We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.

Optimal design of turbines with an attached mass

Boris P. Belinskiy, C. Maeve McCarthy, Terry J. Walters (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We minimize, with respect to shape, the moment of inertia of a turbine having the given lowest eigenfrequency of the torsional oscillations. The necessary conditions of optimality in conjunction with certain physical parameters admit a unique optimal design.

Optimal potentials for Schrödinger operators

Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini, Bozhidar Velichkov (2014)

Journal de l’École polytechnique — Mathématiques

We consider the Schrödinger operator - Δ + V ( x ) on H 0 1 ( Ω ) , where Ω is a given domain of d . Our goal is to study some optimization problems where an optimal potential V 0 has to be determined in some suitable admissible classes and for some suitable optimization criteria, like the energy or the Dirichlet eigenvalues.

Shape optimization problems for metric graphs

Giuseppe Buttazzo, Berardo Ruffini, Bozhidar Velichkov (2014)

ESAIM: Control, Optimisation and Calculus of Variations

Γ):Γ ∈ 𝒜, ℋ1(Γ) = l}, where ℋ1D1,...,Dk }  ⊂ Rd . The cost functional ℰ(Γ) is the Dirichlet energy of Γ defined through the Sobolev functions on Γ vanishing on the points Di. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.

Some new problems in spectral optimization

Giuseppe Buttazzo, Bozhidar Velichkov (2014)

Banach Center Publications

We present some new problems in spectral optimization. The first one consists in determining the best domain for the Dirichlet energy (or for the first eigenvalue) of the metric Laplacian, and we consider in particular Riemannian or Finsler manifolds, Carnot-Carathéodory spaces, Gaussian spaces. The second one deals with the optimal shape of a graph when the minimization cost is of spectral type. The third one is the optimization problem for a Schrödinger potential in suitable classes.

Spectral analysis in a thin domain with periodically oscillating characteristics

Rita Ferreira, Luísa M. Mascarenhas, Andrey Piatnitski (2012)

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). We consider all three cases.

Spectral analysis in a thin domain with periodically oscillating characteristics

Rita Ferreira, Luísa M. Mascarenhas, Andrey Piatnitski (2012)

ESAIM: Control, Optimisation and Calculus of Variations

The paper deals with a Dirichlet spectral problem for an elliptic operator with ε-periodic coefficients in a 3D bounded domain of small thickness δ. We study the asymptotic behavior of the spectrum as ε and δ tend to zero. This asymptotic behavior depends crucially on whether ε and δ are of the same order (δ ≈ ε), or ε is much less than δ(δ = ετ, τ < 1), or ε is much greater than δ(δ = ετ, τ > 1). ...

Spectrum of the Laplacian in narrow tubular neighbourhoods of hypersurfaces with combined Dirichlet and Neumann boundary conditions

David Krejčiřík (2014)

Mathematica Bohemica

We consider the Laplacian in a domain squeezed between two parallel hypersurfaces in Euclidean spaces of any dimension, subject to Dirichlet boundary conditions on one of the hypersurfaces and Neumann boundary conditions on the other. We derive two-term asymptotics for eigenvalues in the limit when the distance between the hypersurfaces tends to zero. The asymptotics are uniform and local in the sense that the coefficients depend only on the extremal points where the ratio of the area of the Neumann...

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