On the topology of surfaces with a common boundary and close DN-maps

D. V. Korikov

Zapiski Nauchnykh Seminarov POMI, Mathematical problems in the theory of wave propagation. Part 51, Tome 506 (2021), pp. 79-88

Résumé

Let $\Omega$ be a smooth compact Riemann surface with the boundary $\Gamma$, аnd $\Lambda: \ H^{1}(\Gamma)\mapsto L_{2}(\Gamma)$, $\Lambda f:=\partial_{\nu}u|_{\Gamma}$ its DN-map, where $u$ obeys $\Delta_{g}u=0$ in $\Omega$ and $u=f$ on $\Gamma$. As is known [1], the genus $m$ of the surface $\Omega$ is determined by its DN-map $\Lambda$. In this article, we prove the existence of Riemann surfaces of arbitrary genus $m'>m$, with boundary $\Gamma$, and such that their DN-maps are arbitrarily close to $\Lambda$ with respect to the operator norm. In other words, an arbitrarily small perturbation of the DN-map may change the surface topology.