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D. V. Korikov

Publications and source records attributed to D. V. Korikov.

7 recordsLinked to original sources
arXiv2026

N-transform and factorization of the DN-map

Let $(Ω,g)$ be a smooth compact 3D Riemannian manifold with the smooth boundary $Γ$, $τ(x):={\rm dist\,}(x,Γ)$, $x\inΩ$; $Ω^τ:=\{x\inΩ\,|\,\,{\rm dist\,}(x,Γ)<τ$\}, $Γ^τ:=\{x\inΩ\,|\,\,{\rm dist\,}(x,Γ)=τ$\}, $τ\geqslant 0$. For the sake of technical simplicity, we deal with $Ω$ diffeomorphic to a ball in $\Bbb R^3$. Let $\mathscr P:=\{\nabla p\,|\,\,p\in H^1(Ω)\}$ be the space of the potential vector fields, and let $\mathscr L_λ:=\{\varkappa\nablaτ\,|\,\,\varkappa\in L_2(Ω)\}$ be the space of the vector fields parallel to $\nablaτ$. The N-transform is a map from $\mathscr P$ to $\mathscr L_λ$ defined layer-wise (in accordance with $Ω=\cup_{τ\geqslant 0}Γ^τ$) by $$ Nh\,\big|_{Γ^τ}:=(P^τh)\big|_{Γ^{τ-0}}, \qquadτ>0, $$ where $P^τ$ are the projections in $\mathscr P$ onto the subspaces $\mathscr P^τ:=\{h\in\mathscr P\,|\,\,{\rm supp\,}h\subset\overline{Ω^τ}\}$. We show that $N$ is a unitary operator. Let $p=p^f(x)$ be a solution to the Dirichlet problem: $Δ_g p=0$ in $Ω\setminusΓ$, $p=f$ on $Γ$. The DN-map $Λ$ is defined by $Λf:=-\langle\nabla p^f,\nablaτ\rangle$ on $Γ$. We show that the N-transform provides a certain factorization $Λ^{-1}=V^*V$ and discuss its possible usefulness for determination of $(Ω,g)$ from $Λ$.

math-ph↗
arXiv2026

On non-homeomorphic surfaces with close DN maps

Let $(M,g)$ be a genus $m$ surface with boundary $Γ$ and DN map $Λ$. Introduce the Schottky double $2M$ of $(M,g)$ and denote by $Sys(2M)$ the length of the shortest closed geodesics in the hyperbolic metrics on $2M$. We prove that $Sys(2M)$ is small if $Λ$ is close, in the operator norm, to the DN map $Λ_*$ of some surface $(M_*,g_*)$ of lower genus $m_*<m$ with the same boundary $Γ$: $$\|Λ-Λ_*\|_{B(H^{1/2}(Γ);H^{-1/2}(Γ))}\to 0\,\Longrightarrow \ Sys(2M)\to 0.$$

math.CV↗
arXiv2022

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

Suppose that $M$ is a Riemann surface with boundary $\partial M$, $Λ$ is its DN-map, and $\mathscr E:M\to\mathbb{C}^{n}$ % $\mathfrak{J}_{M}$ is a holomorphic immersion. Let $M'$ be diffeomorphic to $M$, $\partial M=\partial M'$; let $Λ'$ be the DN map of $M'$. Let us write $M'\in\mathbb M_t$ if $\parallelΛ'-Λ\parallel_{H^{1}(\partial M)\to L_{2}(\partial M)}\leqslant t$ holds. We show that, for any holomorphic immersion $\mathscr{E}: M \to \mathbb C^n$ ($n\geqslant 1$), the relation \begin{equation*} \sup_{M'\in \mathbb{M}_{t}}\inf_{\mathscr{E}'}d_{H}(\mathscr E'(M'),\mathscr{E}(M))\underset{t\to 0}{\longrightarrow}0, \end{equation*} holds, where $d_H$ is the Haussdorf distance in $\mathbb C^n$ and the infimum is taken over all holomorphic immersions $\mathscr E': M'\mapsto\mathbb C^n$.

math-ph↗
arXiv2021

Electric Impedance Tomography problem for surfaces with internal holes

Let $(M,g)$ be a smooth compact Riemann surface with the multicomponent boundary $Γ=Γ_0\cupΓ_1\cup\dots\cupΓ_m=:Γ_0\cup\tildeΓ$. Let $u=u^f$ obey $Δu=0$ in $M$, $u|_{Γ_0}=f,\,\,u|_{\tildeΓ}=0$ (the grounded holes) and $v=v^h$ obey $Δv=0$ in $M$, $v|_{Γ_0}=h,\,\,\partial_νv|_{\tildeΓ}=0$ (the isolated holes). Let $Λ_{g}^{\rm gr}: f\mapsto\partial_νu^f|_{Γ_{0}}$ and $Λ_{g}^{\rm is}: h\mapsto\partial_νv^h|_{Γ_{0}}$ be the corresponding DN-maps. The EIT problem is to determine $M$ from $Λ_{g}^{\rm gr}$ or $Λ_{g}^{\rm is}$. To solve it, an algebraic version of the BC-method is applied. The main instrument is the algebra of holomorphic functions on the ma\-ni\-fold ${\mathbb M}$, which is obtained by gluing two examples of $M$ along $\tildeΓ$. We show that this algebra is determined by $Λ_{g}^{\rm gr}$ (or $Λ_{g}^{\rm is}$) up to isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g',Γ_{0}')$ of $(M,g,Γ_{0})$. This copy is conformally equivalent to the original, provides $Γ_{0}'=Γ_{0},\,\,Λ_{g'}^{\rm gr}=Λ_{g}^{\rm gr},\,\,Λ_{g'}^{\rm is}=Λ_{g}^{\rm is}$, and thus solves the problem.

math-ph↗
arXiv2021

On characterization of Dirichlet-to-Neumann map of Riemannian surface with boundary

Let $(M,g)$ be a smooth compact orientable two-dimensional Riemannian manifold ({\it surface}) with a smooth metric tensor $g$ and smooth connected boundary $Γ$. Its {\it DN-map} $Λ_g:{C^\infty}(Γ)\to{C^\infty}(Γ)$ is associated with the (forward) elliptic problem $ Δ_gu=0 \,\,\, {\rm in}\,\,M\setminusΓ,\,\,u=f \,\,\, {\rm on}\,\,\,Γ$, and acts by $ Λ_g f:=\partial_νu^f \,\,\, {\rm on}\,\,\,Γ, $ where $Δ_g$ is the Beltrami-Laplace operator, $u=u^f(x)$ is the solution, $ν$ is the outward normal to $Γ$. The corresponding {\it inverse problem} is to determine the surface $(M,g)$ from its DN-map $Λ_g$. We provide the necessary and sufficient conditions on an operator acting in ${C^\infty}(Γ)$ to be the DN-map of a surface. In contrast to the known conditions by G.Henkin and V.Michel in terms of multidimensional complex analysis, our ones are based on the connections of the inverse problem with commutative Banach algebras.

math.AP↗
arXiv2020

On the EIT problem for nonorientable surfaces

Let $(Ω,g)$ be a smooth compact two-dimensional Riemannian manifold with boundary, $Λ_g: f\mapsto \partial_νu|_{\partialΩ}$ its DN map, where $u$ obeys $Δ_g u=0$ in $Ω$ and $u|_{\partial Ω}=f$. The Electric Impedance Tomography problem is to determine $Ω$ from $Λ_g$. A criterion is proposed that enables one to detect (via $Λ_g$) whether $Ω$ is orientable or not. The algebraic version of the BC-method is applied to solve the EIT problem for the Moebius band. The main instrument is the algebra of holomorphic functions on the double covering ${\mathbb M}$ of $M$, which is determined by $Λ_g$ up to an isometric isomorphism. Its Gelfand spectrum (the set of characters) plays the role of the material for constructing a relevant copy $(M',g')$ of $(M,g)$. This copy is conformally equivalent to the original, provides $\partial M'=\partial M,\,\,Λ_{g'}=Λ_g$, and thus solves the problem.

math-ph↗