Large $|k|$ behavior of complex geometric optics solutions to d-bar problems

TL;DR

This paper analyzes the behavior of complex geometric optics solutions to d-bar problems at large spectral parameter values, providing asymptotic formulas and numerical validation for specific potentials.

Contribution

It establishes convergence rates of solutions for certain potentials and derives explicit leading order asymptotics, including for characteristic functions of convex sets.

Findings

Solutions converge as a geometric series in 1/|k|^{s-1} for potentials in a specific Sobolev space.

The asymptotic behavior is explicitly computed for characteristic functions of convex sets.

Numerical simulations confirm the theoretical asymptotic formulas for the disk example.

Abstract

Complex geometric optics solutions to a system of d-bar equations appearing in the context of electrical impedance tomography and the scattering theory of the integrable Davey-Stewartson II equations are studied for large values of the spectral parameter $k$. For potentials \( q\in \langle \cdot \rangle^{-2} H^{s}(\mathbb{C}) \) for some $s \in]1,2]$, it is shown that the solution converges as the geometric series in $1/|k|^{s-1}$. For potentials $q$ being the characteristic function of a strictly convex open set with smooth boundary, this still holds with $s=3/2$ i.e., with $1/\sqrt{|k|}$ instead of $1/|k|^{s-1}$. The leading order controbutions are computed explicitly. Numerical simulations show the applicability of the asymptotic formulae for the example of the characteristic function of the disk.