An inverse problem for an inhomogeneous conformal Killing field equation

Abstract:

Let $g$ be a $C^{2,\alpha }$ Riemannian metric defined on a bounded domain $\Omega \subset R^2$ with $C^{3,\alpha }$ boundary and let $X$ be a $C^{2,\alpha }$ vector field on $\bar {\Omega }$ satisfying $X|_{\partial \Omega }=0$. We show that if $l$ is a gradient field of a solution $u$ to the equation $\triangle _gu-\bigl \langle \nabla _{g }\sigma , \nabla _gu\bigr \rangle _g=0$ on $\Omega$, then both inner products $\bigl \langle l,X\bigr \rangle _g$ and $\bigl \langle l^\perp ,X\bigr \rangle _g$ are uniquely determined by the restriction of the tensor ${\mathcal L}_X(g)-(e^\sigma \nabla _{g}\cdot (e^{-\sigma }X)) g$ to the gradient field $l$, where ${\mathcal L}_X(g)$ is the Lie derivative of the metric tensor $g$ under the vector field $X$ and $\sigma =log\sqrt {det(g)}$. This work solves a problem related to an inverse boundary value problem for nonlinear elliptic equations.