Determining $W^{1,n}$ conductivities in Calder\'on's problem for $n\geq4$

Abstract

We establish a global uniqueness result at the critical Sobolev regularity for the isotropic Calder\'on problem in all dimensions $n\ge 4$. More precisely, for every $n\ge4$ and every bounded Lipschitz domain $\Omega\subset\mathbb{R}^n$, any real-valued, uniformly elliptic scalar conductivity $\gamma\in W^{1,n}(\Omega)$ is uniquely determined by its full Dirichlet-to-Neumann map. This settles a long-standing open problem in the field. To prove uniqueness at this endpoint, we develop an averaged trace norm method, which combines complex geometrical optics with trace norm estimates averaged over wave orientations.