On stability of determination of Riemann surface from its DN-map

2021/12/29 by Belishev, M. I., Korikov, D. V.
#30F15 #35R30 #46J15 #46J20 #FOS: Physical sciences #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2112.14816

Abstract

Suppose that M is a Riemann surface with boundary ∂ M, Λ is its DN-map, and \mathscr E:M→ℂⁿ % \mathfrakJ_M is a holomorphic immersion. Let M' be diffeomorphic to M, ∂ M=∂ M'; let Λ' be the DN map of M'. Let us write M'∈\mathbb M_t if ∥Λ'-Λ∥_H¹(∂ M)→ L₂(∂ M)\leqslant t holds. We show that, for any holomorphic immersion \mathscrE: M → \mathbb Cⁿ (n\geqslant 1), the relation sup_M'∈ \mathbbM_tinf_\mathscrE'd_H(\mathscr E'(M'),\mathscrE(M))\undersett→ 0\longrightarrow0, holds, where d_H is the Haussdorf distance in \mathbb Cⁿ and the infimum is taken over all holomorphic immersions \mathscr E': M'↦\mathbb Cⁿ.

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