The Calder\'on problem for the infinity Laplacian with large affine boundary data

Abstract

We study the Calder\'on 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 $\Lambda_q\bigl(t(e\cdot x+b)|_{\partial\Omega}\bigr)$, 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.