Steklov and Robin isospectral manifolds

Abstract

We use two of the most fruitful methods for constructing isospectral manifolds, the Sunada method and the torus action method, to construct manifolds whose Dirichletto-Neumann operators are isospectral at all frequencies. The manifolds are also isospectral for the Robin boundary value problem for all choices of Robin parameter. As in the sloshing problem, we can also impose mixed Dirichlet-Neumann conditions on parts of the boundary. Among the examples we exhibit are Steklov isospectral flat surfaces with boundary, planar domains with isospectral sloshing problems, and Steklov isospectral metrics on balls of any dimension greater than 5. In particular, the latter are the first examples of Steklov isospectral manifolds of dimension greater than 2 that have connected boundaries.

Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons

“…In our case, the construction block 𝐾 is the triangle 𝐴𝐵𝐹, the line 𝑎 is the side 𝐴𝐹, and the line 𝑏 is 𝐵𝐹. Note that although [32,Theorem 3.1] is stated for the Laplacian with mixed Dirichlet-Neumann boundary conditions, its proof applies verbatim in our case, see also [19]. □ Using Lemma 5.12 and applying [33,Proposition 1.13] to the isosceles triangle constructed in the lemma, we immediately obtain that .…”

“…It is easily seen that in the case of the equilateral triangle the eigenfunctions are more or less equally distributed on all sides, whereas in the exceptional case in Figure 3 the eigenfunction 𝑢 18 is mostly concentrated on the union of two sides (and not on one side, cf. Remark 1.18 and Corollary 4.32; note that the corresponding quasi-eigenvalue is double), and the eigenfunction 𝑢 19 is mostly concentrated on the hypothenuse.…”

“…We illustrate Proposition 1.15 by showing, in Figures 2 and 3, the numerically computed boundary traces 𝑢 𝑚 | ⨐ 𝑃 for the equilateral triangle  3 from Example 1.12(a 3 ) (all angles are special) and for the right-angled isosceles triangle 𝑇 1 from Example 1.10 (all angles are exceptional); see Subsection 9.1 for details of the numerical procedure. In both cases, we plot two eigenfunctions 𝑢 18 and 𝑢 19 . For the equilateral triangle, these eigenfunctions correspond to the eigenvalues 𝜆 18 ≈ 17.8023 and 𝜆 19 ≈ 19.8968, which in turn correspond to the quasi-eigenvalues 𝜎 18 = 17𝜋 3 and 𝜎 19 = 19𝜋 3 (both of which are in fact double, 𝜎 17 = 𝜎 18 and 𝜎 19 = 𝜎 20 ).…”

“…In both cases, we plot two eigenfunctions 𝑢 18 and 𝑢 19 . For the equilateral triangle, these eigenfunctions correspond to the eigenvalues 𝜆 18 ≈ 17.8023 and 𝜆 19 ≈ 19.8968, which in turn correspond to the quasi-eigenvalues 𝜎 18 = 17𝜋 3 and 𝜎 19 = 19𝜋 3 (both of which are in fact double, 𝜎 17 = 𝜎 18 and 𝜎 19 = 𝜎 20 ). For the right-angled isosceles triangle, these eigenfunctions correspond to the eigenvalues 𝜆 18 ≈ 15.708 and 𝜆 19 ≈ 16.6608, which in turn correspond to the quasi-eigenvalues 𝜎 18 = 5𝜋 (which is in fact double, 𝜎 17 = 𝜎 18 ) and…”

“…In our case, the construction block $$K$$ is the triangle $$ABF$$, the line $$a$$ is the side $$AF$$, and the line $$b$$ is $$BF$$. Note that although [32, Theorem 3.1] is stated for the Laplacian with mixed Dirichlet–Neumann boundary conditions, its proof applies verbatim in our case, see also [19]. $$\Box$$…”