TITLE: Progress on isospectral plane domains
QUESTION [7 upvotes]: 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.

REPLY [4 votes]: 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.