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Despite the moduli space of triangles being three dimensional, we prove the existence of two triangles which are not isometric to each other for which the first, second and fourth Dirichlet eigenvalues coincide, establishing a numerical observation from Antunes-Freitas [P. R. S. Antunes and P. Freitas. Proc. R. Soc. Lond.Ser. A Math. Phys. Eng. Sci., 467(2130):1546-1562, 2011]. The two triangles are far from any known, explicit cases. To do so, we develop new tools to rigorously enclose eigenvalues to a very high precision, as well as their position in the spectrum. This result is also mentioned as (the negative part of) Conjecture 6.46 in [R. Laugesen, B. Siudeja, Shape optimization and spectral theory, 149-200. De Gruyter Open, Warsaw, 2017], Open Problem 1 in [D. Grieser, S. Maronna, Notices Amer. Math. Soc., 60(11):1440-1447, 2013] and Conjecture 3 in [Z. Lu, J. Rowlett. Amer. Math. Monthly, 122(9):815-835, 2015.].
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This paper is devoted to studying impedance eigenvalues (that is, eigenvalues of a particular Dirichlet-to-Neumann map) for the time harmonic linear elastic wave problem, and their potential use as target-signatures for fluid-solid interaction problems. We first consider several possible families of eigenvalues of the elasticity problem, focusing on certain impedance eigenvalues that are an analogue of Steklov eigenvalues. We show that one of these families arises naturally in inverse scattering. We also analyse their approximation from far field measurements of the scattered pressure field in the fluid, and illustrate several alternative methods of approximation in the case of an isotropic elastic disk.
We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $partial_{ u} u - gamma(x) u = 0$ on the boundary $Gamma$ and $gamma(x) > 0.$ The solutions are described by a contraction semigroup $V(t) = e^{tG}, : t geq 0.$ The eigenvalues $lambda_k$ of $G$ with ${rm Re}: lambda_k < 0$ yield asymptotically disappearing solutions $u(t, x) = e^{lambda_k t} f(x)$ having exponentially decreasing global energy. We establish a Weyl formula for these eigenvalues in the case $min_{xin Gamma} gamma(x) > 1.$ For strictly convex obstacles $K$ this formula concerns all eigenvalues of $G.$
We investigate multiplicity and symmetry properties of higher eigenvalues and eigenfunctions of the $p$-Laplacian under homogeneous Dirichlet boundary conditions on certain symmetric domains $Omega subset mathbb{R}^N$. By means of topological arguments, we show how symmetries of $Omega$ help to construct subsets of $W_0^{1,p}(Omega)$ with suitably high Krasnoselskiu{i} genus. In particular, if $Omega$ is a ball $B subset mathbb{R}^N$, we obtain the following chain of inequalities: $$ lambda_2(p;B) leq dots leq lambda_{N+1}(p;B) leq lambda_ominus(p;B). $$ Here $lambda_i(p;B)$ are variational eigenvalues of the $p$-Laplacian on $B$, and $lambda_ominus(p;B)$ is the eigenvalue which has an associated eigenfunction whose nodal set is an equatorial section of $B$. If $lambda_2(p;B)=lambda_ominus(p;B)$, as it holds true for $p=2$, the result implies that the multiplicity of the second eigenvalue is at least $N$. In the case $N=2$, we can deduce that any third eigenfunction of the $p$-Laplacian on a disc is nonradial. The case of other symmetric domains and the limit cases $p=1$, $p=infty$ are also considered.
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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