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Register a new userExtremum problems for eigenvalues of discrete Laplace operators
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Authors
Ren Guo
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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.
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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 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.
We study the eigenvalues of a Laplace-Beltrami operator defined on the set of the symmetric polynomials, where the eigenvalues are expressed in terms of partitions of integers. By assigning partitions with the restricted uniform measure, the restricted Jack measure, the uniform measure or the Plancherel measure, we prove that the global distribution of the eigenvalues is asymptotically a new distribution $mu$, the Gamma distribution, the Gumbel distribution and the Tracy-Widom distribution, respectively. An explicit representation of $mu$ is obtained by a function of independent random variables. We also derive an independent result on random partitions itself: a law of large numbers for the restricted uniform measure. Two open problems are also asked.
We consider general second order uniformly elliptic operators subject to homogeneous boundary conditions on open sets $phi (Omega)$ parametrized by Lipschitz homeomorphisms $phi $ defined on a fixed reference domain $Omega$. Given two open sets $phi (Omega)$, $tilde phi (Omega)$ we estimate the variation of resolvents, eigenvalues and eigenfunctions via the Sobolev norm $|tilde phi -phi |_{W^{1,p}(Omega)}$ for finite values of $p$, under natural summability conditions on eigenfunctions and their gradients. We prove that such conditions are satisfied for a wide class of operators and open sets, including open sets with Lipschitz continuous boundaries. We apply these estimates to control the variation of the eigenvalues and eigenfunctions via the measure of the symmetric difference of the open sets. We also discuss an application to the stability of solutions to the Poisson problem.
We study spectral instability of steady states to the linearized 2D Euler equations on the torus written in vorticity form via certain Birman-Schwinger type operators $K_{lambda}(mu)$ and their associated 2-modified perturbation determinants $mathcal D(lambda,mu)$. Our main result characterizes the existence of an unstable eigenvalue to the linearized vorticity operator $L_{rm vor}$ in terms of zeros of the 2-modified Fredholm determinant $mathcal D(lambda,0)=det_{2}(I-K_{lambda}(0))$ associated with the Hilbert Schmidt operator $K_{lambda}(mu)$ for $mu=0$. As a consequence, we are also able to provide an alternative proof to an instability theorem first proved by Zhiwu Lin which relates existence of an unstable eigenvalue for $L_{rm vor}$ to the number of negative eigenvalues of a limiting elliptic dispersion operator $A_{0}$.
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