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.

Bibliographie

[1] M. I. Belishev, “The Calderon problem for two-dimensional manifolds by the BC-method”, SIAM Journal of Mathematical Analysis, 35:1 (2003), 172–182 | DOI | Zbl

[2] M. Lassas, G. Uhlmann, “On determining a Riemannian manifold from the Dirichlet-to-Neumann map”, Ann. Scient. Ec. Norm. Sup., 34:5 (2001), 771–787 | DOI | Zbl

[3] V. Maz'ya, S. Nazarov, B. Plamenevskij, Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, v. I, Operator Theory: Advances and Applications, 111, Birkhäuser, Basel, 2000 | Zbl