Uniqueness in the Calderón problem and bilinear restriction estimates (1903.09382v3)

Abstract: Uniqueness in the Calder\'on problem in dimension bigger than two was usually studied under the assumption that conductivity has bounded gradient. For conductivities with unbounded gradients uniqueness results have not been known until recent years. The latest result due to Haberman basically relies on the optimal $L^2$ restriction estimate for hypersurface which is known as the Tomas-Stein restriction theorem. In the course of developments of the Fourier restriction problem bilinear and multilinear generalizations of the (adjoint) restriction estimates under suitable transversality condition between surfaces have played important roles. Since such advanced machineries usually provide strengthened estimates, it seems natural to attempt to utilize these estimates to improve the known results. In this paper, we make use of the sharp bilinear restriction estimates, which is due to Tao, and relax the regularity assumption on conductivity. We also consider the inverse problem for the Schr\"odinger operator with potentials contained in the Sobolev spaces of negative orders.