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Inverse problems for non-linear partial differential equations

Professor Matti LASSAS
Date & Time
24 Feb 2022 (Thu) | 04:00 PM - 05:00 PM
Venue
Online Zoom
Youtube Link

ABSTRACT

In the talk we give an overview on how inverse problems can be used solved using non-linear interaction of the solutions. This method can be used for several different inverse problems for nonlinear hyperbolic or elliptic equations. In this approach one does not consider the non-linearity as a troublesome perturbation term, but as an effect that aids in solving the problem. Using it, one can solve inverse problems for non-linear equations for which the corresponding problem for linear equations is still unsolved. For the hyperbolic equations, we consider the non-linear wave equation gu+um=f\square_g u+u^m=f on a Lorentzian manifold M×RM\times R and the source-to-solution map ΛV:fuV\Lambda_V:f\to u|_V that maps a source ff, supported in an open domain VM×RV\subset M\times R, to the restriction of uu in VV. Under suitable conditions, we show that the observations in VV, that is, the map ΛV\Lambda_V, determine the metric gg in a larger domain which is the maximal domain where signals sent from VV can propagate and return back to VV. We apply non-linear interaction of solutions of the linearized equation also to study non-linear elliptic equations. For example, we consider Δgu+qum=0\Delta_g u+qu^m=0 in ΩRn\Omega\subset R^n with the boundary condition uΩ=fu|_{\partial \Omega}=f. For this equation we define the Dirichlet-to-Neumann map ΛΩ:fuV\Lambda_{\partial \Omega}:f\to u|_V. Using the high-order interaction of the solutions, we consider various inverse problems for the metric gg and the potential qq.

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