Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains

  title={Steklov Eigenvalues and Quasiconformal Maps of Simply Connected Planar Domains},
  author={Alexandre Girouard and Richard S. Laugesen and Bartlomiej A. Siudeja},
  journal={Archive for Rational Mechanics and Analysis},
We investigate isoperimetric upper bounds for sums of consecutive Steklov eigenvalues of planar domains. The normalization involves the perimeter and scale-invariant geometric factors which measure deviation of the domain from roundness. We prove sharp upper bounds for both starlike and simply connected domains for a large collection of spectral functionals including partial sums of the zeta function and heat trace. The proofs rely on a special class of quasiconformal mappings. 

Figures and Tables from this paper

An inequality for the Steklov spectral zeta function of a planar domain
We consider the zeta function  for the Dirichlet–to–Neumann operator of a simply connected planar domain  bounded by a smooth closed curve. We prove that, for a xed real s satisfying jsj > 1 andExpand
An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space
In this article, we prove an isoperimetric inequality for the harmonic mean of the first $(n-1)$ nonzero Steklov eigenvalues on bounded domains in $n$-dimensional Hyperbolic space. Our approach toExpand
Sharp bounds for Steklov eigenvalues on star-shaped domains
In this article, we consider Steklov eigenvalue problem on star-shaped bounded domain Ω in hypersurface of revolution and paraboloid, P = {(x, y, z) ∈ ℝ3 : z = x2 + y2}. A sharp lower bound isExpand
Steklov Eigenvalue Problem on Subgraphs of Integer Lattices
We study the eigenvalues of the Dirichlet-to-Neumann operator on a finite subgraph of the integer lattice Zn. We estimate the first n+1 eigenvalues using the number of vertices of the subgraph. As aExpand
The Steklov problem is an eigenvalue problem with the spectral parameter in the boundary conditions, which has various applications. Its spectrum coincides with that of the Dirichlet-to-NeumannExpand
Maximal Convex Combinations of Sequential Steklov Eigenvalues
The existence of the optimal domain and the nondecreasing, Lipschitz continuity, and convexity of the ideal objective function with respect to the convex combination constant are shown. Expand
Computation of free boundary minimal surfaces via extremal Steklov eigenvalue problems
Recently Fraser and Schoen showed that the solution of a certain extremal Steklov eigenvalue problem on a compact surface with boundary can be used to generate a free boundary minimal surface, i.e.,Expand
Dirichlet-to-Neumann or Poincaré-Steklov operator on fractals described by d -sets
In the framework of the Laplacian transport, described by a Robin boundary value problem in an exterior domain in $\mathbb{R}^n$, we generalize the definition of the Poincare-Steklov operator toExpand
Some sharp bounds for Steklov eigenvalues.
This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $\Omega$ be a star-shaped bounded domainExpand
Optimization of Steklov-Neumann eigenvalues
An algorithm is developed which uses asymptotic formulas concerning perturbations of the partitioning of the boundary pieces, and some results concerning bounds and examples with regards to the governing problem are displayed. Expand


Upper bounds for Steklov eigenvalues on surfaces
We give explicit isoperimetric upper bounds for all Steklov eigenvalues of a compact orientable surface with boundary, in terms of the genus, the length of the boundary, and the number of boundaryExpand
Sharp spectral bounds on starlike domains
We prove sharp bounds on eigenvalues of the Laplacian that complement the Faber--Krahn and Luttinger inequalities. In particular, we prove that the ball maximizes the first eigenvalue and minimizesExpand
Isoperimetric control of the Steklov spectrum
Let N be a complete Riemannian manifold of dimension n+1 whose Riemannian metric g is conformally equivalent to a metric with non-negative Ricci curvature. The normalized Steklov eigenvalues of aExpand
On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues
We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained inExpand
An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem
Let Ω be a bounded smooth domain in ℝn and let 0 = λ1 ≤ λ2 ≤ … denote the eigenvalues of the Stekloff problem: Δu = 0 in Ω and (∥u)/(∥ν) = λi on ∥Ω. We show that , where denotes the second eigenvalueExpand
A Reilly inequality for the first Steklov eigenvalue
Abstract Let M be a compact submanifold with boundary of a Euclidean space or a Sphere. In this paper, we derive an upper bound for the first non-zero eigenvalue p 1 of Steklov problem on M in termsExpand
Sums of reciprocal Stekloff eigenvalues
Let 0 = λ1 < λ2 ≤ λ3 ≤ … be the Stekloff eigenvalues of a plane domain. The paper is concerned with formulas for ∑∞2λ(–2)j in simply and doubly connected domains. In the simply connected case it isExpand
The first Steklov eigenvalue, conformal geometry, and minimal surfaces
We consider the relationship of the geometry of compact Riemannian manifolds with boundary to the first nonzero eigenvalue σ1 of the Dirichlet-to-Neumann map (Steklov eigenvalue). For surfaces Σ withExpand
An inequality for Steklov eigenvalues for planar domains
Study of the zeta function associated to the Neumann operator on planar domains yields an inequality for Steklov eigenvalues for planar domains.
Shape optimization for low Neumann and Steklov eigenvalues
We give an overview of results on shape optimization for low eigenvalues of the Laplacian on bounded planar domains with Neumann and Steklov boundary conditions. These results share a common feature:Expand