The Calderón problem for the infinity Laplacian with large affine boundary data

Abstract

We study the Calderón problem for the equation $-{\Delta }_{\infty }u+q(x)u=0$ in a bounded convex domain with $C^2$ boundary. We prove that a positive potential $q$ is uniquely determined and explicitly reconstructible from the measurements ${\Lambda }_q(t(e\cdot x+b){|}_{\partial \Omega })$, where $e\in {\mathbb{S}}^{n-1}$, the offset $b>{\sup }_{\overline{\Omega }}|x|$ is fixed, and $t\to \infty$. The first potential-dependent term in the large amplitude asymptotics determines weighted integrals of $q$ over the chords parallel to $e$. Combining the measurements in the directions $e$ and $-e$ gives the X-ray transform of the zero extension of $q$, and the Fourier slice identity yields reconstruction and uniqueness.

21 pages