Robin spectral rigidity of nearly circular domains with a reflectional symmetry

Abstract

This is a note on a recent paper of De Simoi -Kaloshin -Wei [DKW16]. We show that using their results combined with wave trace invariants of Guillemin-Melrose [GM79b] and the heat trace invariants of Zayed [Za98] for the Laplacian with Robin boundary conditions, one can extend the Dirichlet/Neumann spectral rigidity results of [DKW16] to the case of Robin boundary conditions. We will consider the same generic subset as in [DKW16] of smooth strictly convex Z 2 -symmetric planar domains sufficiently close to a circle, however we pair them with arbitrary Z 2 -symmetric smooth Robin functions on the boundary and of course allow deformations of Robin functions as well.

Dynamical spectral rigidity among $\mathbb{Z}_2$-symmetric strictly convex domains close to a circle (Appendix B coauthored with H. Hezari)

“…Remark 2.11. Hezari, in a recent preprint (see [11]), using the method of this paper combined with wave trace invariants of Guillemin-Melrose and the heat trace invariants of Zayed for the Laplacian with Robin boundary conditions, show that one can generalize the Dirichlet/Neumann spectral rigidity claimed in the above corollary to the case of Robin boundary conditions (see [11] for the references).…”

“…Consider a periodic orbit of period q and let p ∈ Z denote its winding number. 11 Then we define the rotation number of the orbits as the ratio p/q. The following lemma is a simple consequence of the fact that Ω has Z 2 -symmetry.…”

“…The original version of this paper (arXiv:1606.00230v1) contained an error which was found and corrected by H. Hezari (see also [11]). This resulted in modifying the form of the operator T, statement and proof of Lemma B.1, estimates in Section 5 in the proof of Theorem 4.9.…”

Robin Spectral Rigidity of the Ellipse

“…Combining the results in [dSKW17] and [PS92], it then follows that Z 2 symmetric domains close to a circle are ∆ spectrally rigid amongst a generic class of symmetric domains. In [Hez17], these results are extended to the Robin Laplacian, where the Robin function on the boundary is also allowed to deform through smooth functions with the same same Z 2 symmetry. Our problem is similar in nature but considers ellipses of arbitrary eccentricity, which might not be close to a circle.…”