On the crack inverse problem for pressure waves in half-space

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• Darko Volkov (Worcester Polytechnic Institute)
• Friday 03 February 2023, 09:00-09:45
• Seminar Room 1, Newton Institute.

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RNTW01 - Rich and Nonlinear Tomography (RNT) in Radar, Astronomy and Geophysics

Starting from the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing term on that crack from overdetermined boundary data on a relatively open set of the top plane. This inverse problem is not uniquely solvable unless some additional assumption is made. However, we show that we can differentiate two cracks $\Gamma_1$ and $\Gamma_2$ under the assumption that $\RR^{3 \sm \ov{\Gamma_1\cup \Gamma_2}$ is connected. As we only assume $L}\infty$ regularity for the wavenumber, proving uniqueness for the inverse problem in that case requires using an advanced unique continuation result obtained by Barcelo et al., 1988. In particular, this unique continuation result implies that a solution to the pressure wave equation $(\Delta + k^{2) u =0$ in an open set of $\RR}n$ satisfies the unique continuation property if $k^{2$ is in $L}s_{loc}(\RR^{n)$ with $s>\f{n}{2}$.\\ If $\RR}3 \sm \ov{\Gamma_1\cup \Gamma_2}$ is not connected we provide counterexamples that demonstrate non-uniqueness for the crack inverse problem, even if $\Gamma_1$ and $\Gamma_2$ are smooth and “almost” flat. This requires using arguments borrowed from the analysis of elliptic PDEs on domains with cusps to verify that certain solutions can be extended on the outside of these domains.\\ Finally, we show in the case where $\RR^3 \sm \ov{\Gamma_1\cup \Gamma_2}$ is not necessarily connected that after excluding a discrete set of frequencies, $\Gamma_1$ and $\Gamma_2$ can again be differentiated from overdetermined boundary data.\\

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