Seminar

We consider inverse problems for non-linear wave equations $\square_g u+au^2=f$ in the the space-time $M\times \R$ having the Lorentzian metric $g$. We consider observations in an open subset $V\subset M\times \R$ that is a

neighbourhood of a time-like curve $\mu$. Also, we consider solutions $u(x,t)$ of the wave equation that are produced by sources $f$ supported in $V$ and assume that these solutions are observed in the same set $V$, that is, we observe $u|_V$. We show that the conformal class of the metric $g$ can be determined in a large domain $W\subset M$ which is the maximal domain wherev signals sent from $V$ can propagate and return back to $V$.

The inverse problem for the metric $g$ is solved using the method of artificial microlocal point sources. In this method we do not consider the non-linearity as a troublesome perturbation term, but as an effect that aids in solving the problem. The artificial microlocal point sources are created by distorted plane waves the interact non-linearly and produce singularities that are similar to the singularities produced by a point source that would be in the unknown part $W$ of the space time. Note that as the metric is unknown, the non-linear interaction of the waves need to studied without knowing the coefficients of the wave equation. Using thismethod we are able to solve inverse problems for non-linear equations for which the corresponding problem for linear equations is still unsolved.

The results have been done in collaboration with Xi Chen, Yaroslav Kurylev, Lauri

Oksanen, Gabriel Paternain, Gunther Uhlmann, and Yiran Wang.

References:

[1] Y. Kurylev, M. Lassas, G. Uhlmann: Inverse problems for Lorentzian manifolds

and non-linear hyperbolic equations. Inventiones Mathematicae 212 (2018), no. 3,

781-857

[2] M. Lassas: Inverse problems for linear and non-linear hyperbolic equations.

Proceedings of International Congress of Mathematicians ICM 2018, Rio de

Janeiro, Brazil. 2018, Vol III, pp. 3739-3760.

[3] M. Lassas, G. Uhlmann, Y. Wang: Inverse problems for semilinear wave

equations on Lorentzian manifolds. Communications in Mathematical Physics 360

(2018), 555-609.

[4] X. Chen, M. Lassas, L. Oksanen, G. Paternain: Detection of Hermitian

connections in wave equations with cubic non-linearity, arXiv:1902.05711.

[5] Y. Kurylev, M. Lassas, L. Oksanen, G. Uhlmann: Inverse problem for Einstein-

scalar field equations, arXiv:1406.4776.