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Has there been any progress on the smooth isospectral plane domains for Laplacian problem with Dirichlet data? In particular, are there known examples of domains which are isospectral to the unit disk?

Related to this and of course this.

Edit 1: What if a non-local boundary condition is enforced? To be precise, suppose $D$ is an open disk, and $\Omega$ is a symmetric bounded domain with smooth boundary so that $D$ and $\Omega$ are isospectral w.r.t to Laplacian with non-local boundary condition. Does it follow that $D$ and $\Omega$ are congruent?

Update. Z.Lu and J.Rowlett [paper] recently proved the following:

Theorem. Let Ω be a simply connected planar domain with piecewise smooth Lipschitz boundary. If Ω has at least one corner, then Ω is not isospectral to any bounded planar domain with smooth boundary that has no corners.

Corollary. Amongst all planar domains of fixed genus with piecewise smooth Lipschitz boundary, those that have at least one corner are spectrally distinguished.

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  • $\begingroup$ What do you mean by "non-local boundary condition"? $\endgroup$ – Otis Chodosh Nov 26 '15 at 8:23
  • $\begingroup$ @OtisChodosh like some integral jump the one Im interested in is as follows: $u(x)+\int_{\partial\Omega}u(y)\partial_{n}\ln|x−y|ds_y−\int_{\partial\Omega} \frac{\partial u(y)}{\partial n^+} \ln|x−y|ds_y=0$ for $x\in\partial\Omega$ $\endgroup$ – BigM Nov 26 '15 at 16:46
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    $\begingroup$ The disc is determined by its spectrum: it is determined by its area and perimeter (isoperimetric inequality), and the area and perimeter are spectral invariants (you can hear the area and the perimeter of a drum, cf. the comments on <mathoverflow.net/questions/245180>). $\endgroup$ – Noam D. Elkies Nov 18 '16 at 4:37
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The Dirichlet/Neumann spectrum determines the domain among the class of analytic domains with some discrete symmetries by work of Zelditch.

The unit disk is determined by its Dirichlet spectrum (among any region, with sufficiently regular boundary; I'm not sure what the minimal assumptions are): First, by Weyl's law, "you can hear the area of a drum" (i.e., the area is a spectral invariant). Then, the claim follows from the rigidity statement in the Faber--Krahn inequality.

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  • $\begingroup$ at Otis Chodosh, I went back and glanced at Steve Zelditch's paper. The classes he considers are pretty "large" and include a lot of nice domains. Do you know of studies considering other appropriate boundary conditions e.g. Robin or mixed? My understanding, as an outsider to the business of isospectral geometry, is that boundary conditions often come in as variational expression for eigenvalues and can drastically change the scenario. $\endgroup$ – BigM Oct 4 '16 at 16:30
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    $\begingroup$ @BigM, as far as i know, there are no results along the lines of Zelditch for mixed boundary conditions. This paper: arxiv.org/pdf/math/0510505v2.pdf gives some constructions of isospectral domains with piecewise smooth boundary. $\endgroup$ – Otis Chodosh Oct 7 '16 at 12:34

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