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The exterior Steklov problem for Euclidean domains

Lukas Bundrock, Alexandre Girouard, Denis S. Grebenkov, Michael Levitin, Iosif Polterovich

TL;DR

<3-5 sentence high-level summary> The exterior Steklov problem studies harmonic functions in the exterior of a bounded domain with a boundary condition that links the normal derivative to the boundary values. The paper unifies multiple formulations (finite-energy, conformal in 2D, truncated domains, Helmholtz, and layer potentials) and proves their equivalence, enabling robust spectral analysis. It establishes an Escobar-type lower bound for the first exterior eigenvalue in dimensions n≥3, sharp for balls, and shows that fixed-volume convex domains can have first exterior eigenvalues diverging to infinity, contrasting with interior and 2D exterior behavior where isoperimetric-type bounds persist. In 2D, a Weinstock-type isoperimetric inequality is proved for the exterior problem, while in higher dimensions no such bound holds. The work also develops Weyl-type asymptotics and detailed properties of eigenfunctions, and provides explicit examples and numerical demonstrations to illustrate the theory.

Abstract

We investigate the Steklov eigenvalue problem in an exterior Euclidean domain. First, we present several formulations of this problem and establish the equivalences between them. Next, we examine various properties of the exterior Steklov eigenvalues and eigenfunctions. One of our main findings is an Escobar-type lower bound for the first exterior Steklov eigenvalue on convex domains in dimensions three and higher. This bound is expressed in terms of the principal curvatures of the boundary and is sharp, with equality attained for a ball. Moreover, it implies the existence of a sequence of convex domains with fixed volume and the first exterior Steklov eigenvalues tending to infinity. This contrasts with the interior case, as well as with the two-dimensional exterior case, for which we show that an analogue of the Weinstock isoperimetric inequality holds.

The exterior Steklov problem for Euclidean domains

TL;DR

<3-5 sentence high-level summary> The exterior Steklov problem studies harmonic functions in the exterior of a bounded domain with a boundary condition that links the normal derivative to the boundary values. The paper unifies multiple formulations (finite-energy, conformal in 2D, truncated domains, Helmholtz, and layer potentials) and proves their equivalence, enabling robust spectral analysis. It establishes an Escobar-type lower bound for the first exterior eigenvalue in dimensions n≥3, sharp for balls, and shows that fixed-volume convex domains can have first exterior eigenvalues diverging to infinity, contrasting with interior and 2D exterior behavior where isoperimetric-type bounds persist. In 2D, a Weinstock-type isoperimetric inequality is proved for the exterior problem, while in higher dimensions no such bound holds. The work also develops Weyl-type asymptotics and detailed properties of eigenfunctions, and provides explicit examples and numerical demonstrations to illustrate the theory.

Abstract

We investigate the Steklov eigenvalue problem in an exterior Euclidean domain. First, we present several formulations of this problem and establish the equivalences between them. Next, we examine various properties of the exterior Steklov eigenvalues and eigenfunctions. One of our main findings is an Escobar-type lower bound for the first exterior Steklov eigenvalue on convex domains in dimensions three and higher. This bound is expressed in terms of the principal curvatures of the boundary and is sharp, with equality attained for a ball. Moreover, it implies the existence of a sequence of convex domains with fixed volume and the first exterior Steklov eigenvalues tending to infinity. This contrasts with the interior case, as well as with the two-dimensional exterior case, for which we show that an analogue of the Weinstock isoperimetric inequality holds.

Paper Structure

This paper contains 47 sections, 45 theorems, 311 equations, 9 figures, 1 table.

Table of Contents

  1. Introduction and main results
  2. Statement of the problem
  3. Different approaches to the exterior Steklov problem and their equivalence
  4. Spectral properties of the exterior Steklov problem
  5. Courant's theorem and multiplicity bounds
  6. Escobar-type lower bounds for the first eigenvalue
  7. Isoperimetric upper bounds for the first eigenvalue
  8. Exterior Robin problem
  9. Eigenvalue asymptotics
  10. Outline of the paper
  11. Acknowledgements
  12. Examples
  13. Exterior problem for a disk
  14. Exterior problem for Euclidean balls in dimensions $n \ge 3$
  15. Exterior Dirichlet-to-Neumann maps and harmonic extensions for disks and balls
  16. ...and 32 more sections

Key Result

Theorem 1.1

Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with Lipschitz boundary and with connected $\Omega^\mathrm{ext}$. For any $f\in H^{1/2}(\partial\Omega)$, there exists a unique harmonic extension $u=\mathcal{H}^\mathrm{ext} f\in H^1_\mathrm{loc}\left(\Omega^\mathrm{ext}\right)$ satisfying eq:in

Figures (9)

  • Figure 1: The geometry of an exterior problem.
  • Figure 2: Relations between approaches for $n\ge 3$.
  • Figure 3: Relations between approaches for $n=2$.
  • Figure 4: Density plots of eigenfunctions of the Steklov problem in the exterior $\mathcal{K}^\mathrm{ext}$ of the kite.
  • Figure 5: Density plots of eigenfunctions of the Steklov problem in the exterior $\mathcal{T}^\mathrm{ext}$ of the disjoint union of three disks.
  • ...and 4 more figures

Theorems & Definitions (99)

  • Theorem 1.1: see KNP25 and mclean2000strongly
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Theorem 1.8
  • Corollary 1.9
  • Remark 1.10
  • ...and 89 more