N-transform and factorization of the DN-map

Abstract

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