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][1] and of course [this][2].

*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]][3] 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.*


  [1]: https://mathoverflow.net/questions/223672/can-one-hear-the-shape-of-a-drum-for-operators
  [2]: https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/MarkKac.pdf
  [3]: http://www.math.chalmers.se/~rowlett/BLMS-LuR.pdf