N-transform and factorization of the DN-map

Abstract

Let $(\hat{I}\text{c◯},g)$ be a smooth compact 3D Riemannian manifold with the smooth boundary $\hat{I}“$, $\ddot{I}„(x):=\mathrm{dist}(x,\hat{I}“)$, $x\in \hat{I}\text{c◯}$; $\hat{I}{\text{c◯}}^{\ddot{I}}„:=\{x\in \hat{I}\text{c◯}|\mathrm{dist}(x,\hat{I}“)<\ddot{I}„$\}, $\hat{I}{“}^{\ddot{I}}„:=\{x\in \hat{I}\text{c◯}|\mathrm{dist}(x,\hat{I}“)=\ddot{I}„$\}, $\ddot{I}„⩾0$. For the sake of technical simplicity, we deal with $\hat{I}\text{c◯}$ diffeomorphic to a ball in ${\mathbb{R}}^3$. Let $\mathcal{P}:=\{\nabla p|p\in H^1(\hat{I}\text{c◯})\}$ be the space of the potential vector fields, and let ${\mathcal{L}}_{\hat{I}}»:=\{\varkappa \nabla \ddot{I}„|\varkappa \in L_2(\hat{I}\text{c◯})\}$ be the space of the vector fields parallel to $\nabla \ddot{I}„$. The N-transform is a map from $\mathcal{P}$ to ${\mathcal{L}}_{\hat{I}}»$ defined layer-wise (in accordance with $\hat{I}\text{c◯}={\cup }_{\ddot{I}„⩾0}\hat{I}{“}^{\ddot{I}}„$) by $$Nh{|}_{\hat{I}{“}^{\ddot{I}}„}:=(P^{\ddot{I}}„h){|}_{\hat{I}{“}^{\ddot{I}„-0}},\ddot{I}„>0,$$ where $P^{\ddot{I}}„$ are the projections in $\mathcal{P}$ onto the subspaces ${\mathcal{P}}^{\ddot{I}}„:=\{h\in \mathcal{P}|\mathrm{supp}h\subset \overline{\hat{I}{\text{c◯}}^{\ddot{I}}„}\}$. We show that $N$ is a unitary operator. Let $p=p^f(x)$ be a solution to the Dirichlet problem: $\hat{I}{”}_{g}p=0$ in $\hat{I}\text{c◯}\setminus \hat{I}“$, $p=f$ on $\hat{I}“$. The DN-map $\hat{I}›$ is defined by $\hat{I}›f:=-⟨\nabla p^f,\nabla \ddot{I}„⟩$ on $\hat{I}“$. We show that the N-transform provides a certain factorization $\hat{I}{›}^{-1}=V^{\ast }V$ and discuss its possible usefulness for determination of $(\hat{I}\text{c◯},g)$ from $\hat{I}›$.