Do you want to publish a course? Click here

An isoperimetric inequality for Laplace eigenvalues on the sphere

86   0   0.0 ( 0 )
 Added by Alexei V. Penskoi
 Publication date 2017
  fields
and research's language is English




Ask ChatGPT about the research

We show that for any positive integer k, the k-th nonzero eigenvalue of the Laplace-Beltrami operator on the two-dimensional sphere endowed with a Riemannian metric of unit area, is maximized in the limit by a sequence of metrics converging to a union of k touching identical round spheres. This proves a conjecture posed by the second author in 2002 and yields a sharp isoperimetric inequality for all nonzero eigenvalues of the Laplacian on a sphere. Earlier, the result was known only for k=1 (J. Hersch, 1970), k=2 (N. Nadirashvili, 2002; R. Petrides, 2014) and k=3 (N. Nadirashvili and Y. Sire, 2017). In particular, we argue that for any k>=2, the supremum of the k-th nonzero eigenvalue on a sphere of unit area is not attained in the class of Riemannian metrics which are smooth outsitde a finite set of conical singularities. The proof uses certain properties of harmonic maps between spheres, the key new ingredient being a bound on the harmonic degree of a harmonic map into a sphere obtained by N. Ejiri.



rate research

Read More

The paper is concerned with the maximization of Laplace eigenvalues on surfaces of given volume with a Riemannian metric in a fixed conformal class. A significant progress on this problem has been recently achieved by Nadirashvili-Sire and Petrides using related, though different methods. In particular, it was shown that for a given $k$, the maximum of the $k$-th Laplace eigenvalue in a conformal class on a surface is either attained on a metric which is smooth except possibly at a finite number of conical singularities, or it is attained in the limit while a bubble tree is formed on a surface. Geometrically, the bubble tree appearing in this setting can be viewed as a union of touching identical round spheres. We present another proof of this statement, developing the approach proposed by the second author and Y. Sire. As a side result, we provide explicit upper bounds on the topological spectrum of surfaces.
We prove an Herschs type isoperimetric inequality for the third positive eigenvalue on $mathbb S^2$. Our method builds on the theory we developped to construct extremal metrics on Riemannian surfaces in conformal classes for any eigenvalue.
232 - Q. Ding , G. Feng , Y. Zhang 2011
} In this article, we put forward a Neumann eigenvalue problem for the bi-harmonic operator $Delta^2$ on a bounded smooth domain $Om$ in the Euclidean $n$-space ${bf R}^n$ ($nge2$) and then prove that the corresponding first non-zero eigenvalue $Upsilon_1(Om)$ admits the isoperimetric inequality of Szego-Weinberger type: $Upsilon_1(Om)le Upsilon_1(B_{Om})$, where $B_{Om}$ is a ball in ${bf R}^n$ with the same volume of $Om$. The isoperimetric inequality of Szego-Weinberger type for the first nonzero Neumann eigenvalue of the even-multi-Laplacian operators $Delta^{2m}$ ($mge1$) on $Om$ is also exploited.
We prove two explicit bounds for the multiplicities of Steklov eigenvalues $sigma_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given smooth Riemannian surface with boundary, the multiplicities of Steklov eigenvalues $sigma_k$ are uniformly bounded in $k$.
125 - Ren Guo 2011
The discrete Laplace operator on a triangulated polyhedral surface is related to geometric properties of the surface. This paper studies extremum problems for eigenvalues of the discrete Laplace operators. Among all triangles, an equilateral triangle has the maximal first positive eigenvalue. Among all cyclic quadrilateral, a square has the maximal first positive eigenvalue. Among all cyclic $n$-gons, a regular one has the minimal value of the sum of all nontrivial eigenvalues and the minimal value of the product of all nontrivial eigenvalues.
comments
Fetching comments Fetching comments
Sign in to be able to follow your search criteria
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا