N-transform and factorization of the DN-map

Abstract

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 $Λ$.