Lower bounds on Steklov eigenfunctions for a class of curvilinear polygons

Abstract

Steklov eigenfunctions, $\{{\varphi }_{{\sigma }_j}{\}}_{j=0}^{\infty }$, are the eigenfunctions of the Dirichlet to Neumann map on a bounded domain with eigenvalue ${\sigma }_j$. This paper considers Steklov eigenfunctions and eigenvalues on a family of curvilinear polygons, ${\Omega }_{\alpha }$ whose boundaries are smooth away from a single corner of angle $\alpha$. We (i) prove optimal lower bounds on the decay of the harmonic extension, $u_{{\sigma }_j}$, of ${\varphi }_{{\sigma }_j}$ away from the boundary of ${\Omega }_{\alpha }$ and (ii) give optimal lower bounds on the degree to which the high energy Steklov spectrum of ${\Omega }_{\alpha }\setminus H_1$ may differ from that of ${\Omega }_{\alpha }\setminus H_0$ when $H_1$ and $H_0$ are isometric. This shows that the corresponding estimates in recent work by the authors together with Malagutti are optimal. The proof in this paper was obtained through an interaction with ChatGPT 6 Astra: While writing the work with Malagutti, the authors became curious about optimal error bounds for the difference between the Steklov spectra on a curvilinear polygon, $P$, and the spectrum of any operator which knows only local information about $\partial P$. The authors suggested to ChatGPT to remove two isometric, smooth domains from a curvilinear polygon. ChatGPT suggested the ideas for a proof based on a regularized trace and the singularity in the kernel of the DtN map near a corner. The authors then suggested a simpler, conformal approach, the method for handling small angles, and the idea to obtain lower bounds on Steklov eigenfunctions. The paper is entirely written by the authors and the argument has been significantly modified, clarified, and simplified from the ChatGPT ideas.