Inverse boundary value problems for wave equations with quadratic nonlinearities

We study inverse problems for the nonlinear wave equation □gu+w(x,u,∇gu)=0 in a Lorentzian manifold (M,g) with boundary, where ∇gu denotes the gradient and w(x,u,ξ) is smooth and quadratic in ξ. Under appropriate assumptions, we show that the conformal class of the Lorentzian metric g can be recover...

Full description

Saved in:
Bibliographic Details
Published in:Journal of Differential Equations 2022-02, Vol.309, p.558-607
Main Authors: Uhlmann, Gunther, Zhang, Yang
Format: Article
Language:English
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:We study inverse problems for the nonlinear wave equation □gu+w(x,u,∇gu)=0 in a Lorentzian manifold (M,g) with boundary, where ∇gu denotes the gradient and w(x,u,ξ) is smooth and quadratic in ξ. Under appropriate assumptions, we show that the conformal class of the Lorentzian metric g can be recovered up to diffeomorphisms, from the knowledge of the Neumann-to-Dirichlet map. With some additional conditions, we can recover the metric itself up to diffeomorphisms. Moreover, we can recover the second and third quadratic forms in the Taylor expansion of w(x,u,ξ) with respect to u up to null forms.
ISSN:0022-0396
1090-2732
DOI:10.1016/j.jde.2021.11.033