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?
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.