Semilinear Calderón Problem on Stein Manifolds With Kähler Metric

Abstract

We extend existing methods which treat the semilinear Calderón problem on a bounded domain to a class of complex manifolds with Kähler metric. Given two semilinear Schrödinger operators with the same Dirchlet-to-Neumann data, we show that the integral identities that appear naturally in the determination of the analytic potentials are enough to deduce uniqueness on the boundary up to infinite order. By exploiting the assumed complex structure, this information allows us to apply the method of stationary phase and recover the potentials in the interior as well.

A remark on inverse problems for nonlinear magnetic Schrödinger equations on complex manifolds

“…Let us finally mention that inverse problems for the semilinear Schrödinger operators and for nonlinear conductivity equations have been investigated intensively recently, see for example [15], [30], [31], [35], [25], [26], and [9], [21], [7], [8], [40], [43], respectively. Theorem 1.1 is a direct consequence of the main result of [19], combined with some boundary determination results of [38] and of Appendix A, as well as the higher order linearization procedure introduced in [28] in the hyperbolic case, and in [15], [31] in the elliptic case. We refer to [20] where the method of a first order linearization was pioneered in the study of inverse problems for nonlinear PDE, and to [3], [10], [44], and [45] where a second order linearization was successfully exploited.…”

“…Theorem 1.1 in the case of a semilinear Schrödinger operator, i.e. when A = 0, was obtained in [38].…”

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

“…In the case of Schrödinger operator, the constructive determination of the boundary values of a continuous potential from boundary measurements is given in [18, Appendix A], and our reconstruction here will rely crucially on this work. For the nonconstructive boundary determination of a continuous potential in the case of the Schrödinger operator, we refer to the works [23], [36], [39]. Our result is as follows.…”

Reconstructing a potential perturbation of the biharmonic operator on transversally anisotropic manifolds

“…In the case of the Schrödinger operator, the constructive determination of the boundary values of a continuous potential from boundary measurements is given in [19, Appendix A], and our reconstruction here will rely crucially on this work. For the non-constructive boundary determination of a continuous potential in the case of the Schrödinger operator, we refer to the works [26], [38], [42]. For the boundary determination of smooth perturbations based on pseudodifferential techniques, see [40] and [33].…”