570 research outputs found
Dirichlet eigenfunctions on the cube, sharpening the Courant nodal inequality
This paper is devoted to the refine analysis of Courant's theorem for the
Dirichlet Laplacian. Many papers (and some of them quite recent) have
investigated in which cases this inequality in Courant's theorem is an
equality: Pleijel, Helffer--Hoffmann-Ostenhof--Terracini,
Helffer--Hoffmann-Ostenhof, B\'erard-Helffer, Helffer--Persson-Sundqvist,
L\'ena, Leydold. All these results were devoted to -cases in open sets
in or in surfaces like or .
The aim of the current paper is to look for analogous results for domains in
and, as Pleijel was suggesting in his 1956 founding paper,
for the simplest case of the cube. More precisely, we will prove that the only
eigenvalues of the Dirichlet Laplacian which are Courant sharp are the two
first eigenvalues
Extremal first Dirichlet eigenvalue of doubly connected plane domains and dihedral symmetry
We deal with the following eigenvalue optimization problem: Given a bounded
domain , how to place an obstacle of fixed shape within
so as to maximize or minimize the fundamental eigenvalue of the
Dirichlet Laplacian on . This means that we want to extremize the
function , where runs over
the set of rigid motions such that . We answer this problem
in the case where both and are invariant under the action of a dihedral
group , , and where the distance from the origin to the
boundary is monotonous as a function of the argument between two axes of
symmetry. The extremal configurations correspond to the cases where the axes of
symmetry of coincide with those of .Comment: To appear in SIAM Journal on Mathematical Analysi
Artistic Identities and Professional Strategies : Francophone Musicians in France and Britain
Funding This work was supported by Arts and Humanities Research Council [grant number AH/E508628/1] and European Commission [grant number HPSE-CT-2002-00133].Peer reviewedPostprin
Where to place a spherical obstacle so as to maximize the second Dirichlet eigenvalue
We prove that among all doubly connected domains of bounded by
two spheres of given radii, the second eigenvalue of the Dirichlet Laplacian
achieves its maximum when the spheres are concentric (spherical shell). The
corresponding result for the first eigenvalue has been established by Hersch in
dimension 2, and by Harrell, Kr\"oger and Kurata and Kesavan in any dimension.
We also prove that the same result remains valid when the ambient space
is replaced by the standard sphere or the
hyperbolic space .Comment: To appear in Communications in Pure and Applied Analysi
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