44 research outputs found
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
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
Optimal Shapes for the First Dirichlet Eigenvalue of the -Laplacian and Dihedral symmetry
In this paper, we consider the optimization problem for the first Dirichlet
eigenvalue of the -Laplacian , ,
over a family of doubly connected planar domains , where is an open disk and is a domain which
is invariant under the action of a dihedral group for some . We study the behaviour of with respect
to the rotations of about its center. We prove that the extremal
configurations correspond to the cases where is symmetric with respect
to the line containing both the centers. Among these optimizing domains, the
OFF configurations correspond to the minimizing ones while the ON
configurations correspond to the maximizing ones. Furthermore, we obtain
symmetry (periodicity) and monotonicity properties of with respect
to these rotations. In particular, we prove that the conjecture formulated in
[14] for odd and holds true. As a consequence of our monotonicity
results, we show that if the nodal set of a second eigenfunction of the
-Laplacian possesses a dihedral symmetry of the same order as that of ,
then it can not enclose .Comment: 15 pages, 6 figure
On the placement of an obstacle so as to optimize the Dirichlet heat trace
We prove that among all doubly connected domains of bounded by two
spheres of given radii, , the trace of the heat kernel with Dirichlet
boundary conditions, achieves its minimum when the spheres are concentric
(i.e., for the spherical shell). The supremum is attained when the interior
sphere is in contact with the outer sphere.This is shown to be a special case
of a more general theorem characterizing the optimal placement of a spherical
obstacle inside a convex domain so as to maximize or minimize the trace of the
Dirichlet heat kernel. In this case the minimizing position of the center of
the obstacle belongs to the "heart" of the domain, while the maximizing
situation occurs either in the interior of the heart or at a point where the
obstacle is in contact with the outer boundary. Similar statements hold for the
optimal positions of the obstaclefor any spectral property that can be obtained
as a positivity-preserving or positivity-reversing transform of
,including the spectral zeta function and, through it, the regularized
determinant.Comment: in SIAM Journal on Mathematical Analysis, Society for Industrial and
Applied Mathematics, 201
Some recent developments on the Steklov eigenvalue problem
The Steklov eigenvalue problem, first introduced over 125 years ago, has seen
a surge of interest in the past few decades. This article is a tour of some of
the recent developments linking the Steklov eigenvalues and eigenfunctions of
compact Riemannian manifolds to the geometry of the manifolds. Topics include
isoperimetric-type upper and lower bounds on Steklov eigenvalues (first in the
case of surfaces and then in higher dimensions), stability and instability of
eigenvalues under deformations of the Riemannian metric, optimisation of
eigenvalues and connections to free boundary minimal surfaces in balls, inverse
problems and isospectrality, discretisation, and the geometry of
eigenfunctions. We begin with background material and motivating examples for
readers that are new to the subject. Throughout the tour, we frequently compare
and contrast the behavior of the Steklov spectrum with that of the Laplace
spectrum. We include many open problems in this rapidly expanding area.Comment: 157 pages, 7 figures. To appear in Revista Matem\'atica Complutens
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Discrete Differential Geometry
This is the collection of extended abstracts for the 24 lectures and the open problems session at the third Oberwolfach workshop on Discrete Differential Geometry
Two Generator groups acting on the complex hyperbolic plane
This is an expository article about groups generated by two isometries of the
complex hyperbolic plane.Comment: 49 pages, 10 figures. It will appear as a chapter of Volume VI of the
Handbook of Teichmuller theor
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao