2,166 research outputs found
On the minimization of Dirichlet eigenvalues of the Laplace operator
We study the variational problem \inf \{\lambda_k(\Omega): \Omega\
\textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \},
where is the 'th eigenvalue of the Dirichlet Laplacian
acting in , \h(\partial \Omega) is the - dimensional
Hausdorff measure of the boundary of , and is the Lebesgue
measure of . If , and , then there exists a convex
minimiser . If , and if is a minimiser,
then is also a
minimiser, and is connected. Upper bounds are
obtained for the number of components of . It is shown that if
, and then has at most components.
Furthermore is connected in the following cases : (i) (ii) and (iii) and (iv) and
. Finally, upper bounds on the number of components are obtained for
minimisers for other constraints such as the Lebesgue measure and the torsional
rigidity.Comment: 16 page
Minimization of the eigenvalues of the dirichlet-laplacian with a diameter constraint
In this paper we look for the domains minimizing the hth eigenvalue of the Dirichlet-Laplacian λh with a constraint on the diameter. Existence of an optimal domain is easily obtained and is attained at a constant width body. In the case of a simple eigenvalue, we provide nonstandard (i.e., nonlocal) optimality conditions. Then we address the question of whether the disk is an optimal domain in the plane, and we give the precise list of the 17 eigenvalues for which the disk is a local minimum. We conclude by some numerical simulations showing the 20 first optimal domains in the plane
Some isoperimetric inequalities with application to the Stekloff problem
In this paper we establish isoperimetric inequalities for the product of some
moments of inertia. As an application, we obtain an isoperimetric inequality
for the product of the first nonzero eigenvalues of the Stekloff problem in
Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift
This paper deals with the eigenvalue problem for the operator L=-δ-x{dot operator}∇ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue λk of L under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any c>0 and k∈N the following minimization problemmin<>{λk(Ω):Ωquasi-openset,∫Ωe|x|2/2dx≤c} has a solution
Optimization problem for extremals of the trace inequality in domains with holes
We study the Sobolev trace constant for functions defined in a bounded domain
\O that vanish in the subset We find a formula for the first variation
of the Sobolev trace with respect to hole. As a consequence of this formula, we
prove that when \O is a centered ball, the symmetric hole is critical when we
consider deformation that preserve volume but is not optimal for some case.Comment: 13 page
About the connection between the power spectrum of the Cosmic Microwave Background and the Fourier spectrum of rings on the sky
In this article we present and study a scaling law of the CMB
Fourier spectrum on rings which allows us (i) to combine spectra corresponding
to different colatitude angles (e.g. several detectors at the focal plane of a
telescope), and (ii) to recover the power spectrum once the
coefficients have been measured. This recovery is performed numerically below
the 1% level for colatitudes degrees. In addition, taking
advantage of the smoothness of the and of the , we provide
analytical expressions which allow to recover one of the spectrum at the 1%
level, the other one being known.Comment: 8 pages, 8 figure
Maximizing Neumann fundamental tones of triangles
We prove sharp isoperimetric inequalities for Neumann eigenvalues of the
Laplacian on triangular domains.
The first nonzero Neumann eigenvalue is shown to be maximal for the
equilateral triangle among all triangles of given perimeter, and hence among
all triangles of given area. Similar results are proved for the harmonic and
arithmetic means of the first two nonzero eigenvalues
Steklov-type eigenvalues associated with best Sobolev trace constants: domain perturbation and overdetermined systems
We consider a variant of the classic Steklov eigenvalue problem, which arises
in the study of the best trace constant for functions in Sobolev space. We
prove that the elementary symmetric functions of the eigenvalues depend
real-analytically upon variation of the underlying domain and we compute the
corresponding Hadamard-type formulas for the shape derivatives. We also
consider isovolumetric and isoperimetric domain perturbations and we
characterize the corresponding critical domains in terms of appropriate
overdetermined systems. Finally, we prove that balls are critical domains for
the elementary symmetric functions of the eigenvalues subject to volume or
perimeter constraint
Approximation of the critical buckling factor for composite panels
This article is concerned with the approximation of the critical buckling factor for thin composite plates. A new method to improve the approximation of this critical factor is applied based on its behavior with respect to lamination parameters and loading conditions. This method allows accurate approximation of the critical buckling factor for non-orthotropic laminates under complex combined loadings (including shear loading). The influence of the stacking sequence and loading conditions is extensively studied as well as properties of the critical buckling factor behavior (e.g concavity over tensor D or out-of-plane lamination parameters). Moreover, the critical buckling factor is numerically shown to be piecewise linear for orthotropic laminates under combined loading whenever shear remains low and it is also shown to be piecewise continuous in the general case. Based on the numerically observed behavior, a new scheme for the approximation is applied that separates each buckling mode and builds linear, polynomial or rational regressions for each mode. Results of this approach and applications to structural optimization are presented
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