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SUMMARY:On the crack inverse problem for pressure waves in half-space - Da
 rko Volkov (Worcester Polytechnic Institute)
DTSTART:20230203T090000Z
DTEND:20230203T094500Z
UID:TALK194569@talks.cam.ac.uk
DESCRIPTION:Starting from the pressure wave equation in half-space minus a
  crack with a zero Neumann condition on the top plane\, we introduce a rel
 ated inverse problem. That inverse problem consists of identifying the cra
 ck 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 n
 ot uniquely solvable unless some additional assumption is made. However\, 
 we show that we can differentiate two cracks $\\Gamma_1$ and $\\Gamma_2$ u
 nder the assumption that $\\RR^3 \\sm \\ov{\\Gamma_1\\cup \\Gamma_2}$ is c
 onnected. As we only assume $L^\\infty$ regularity for the wavenumber\, pr
 oving uniqueness for the inverse problem in that case requires using an ad
 vanced unique continuation result obtained by Barcelo et al.\, 1988. In pa
 rticular\, this unique continuation result implies that a solution to the 
 pressure wave equation $(\\Delta + k^2) u =0$ in an open set of $\\RR^n$ s
 atisfies the unique continuation property if $k^2$ is in $L^s_{loc}(\\RR^n
 )$ with $s>\\f{n}{2}$.\\\\\nIf $\\RR^3 \\sm \\ov{\\Gamma_1\\cup \\Gamma_2}
 $ is not connected we provide counterexamples that demonstrate non-uniquen
 ess for the crack inverse problem\, even if $\\Gamma_1$ and $\\Gamma_2$ ar
 e smooth and "almost" flat. This requires using arguments borrowed from th
 e analysis of elliptic PDEs on domains with cusps to verify that certain s
 olutions can be extended on the outside of these domains.\\\\\nFinally\, w
 e show in the case where $\\RR^3 \\sm \\ov{\\Gamma_1\\cup \\Gamma_2}$ is n
 ot necessarily connected that after excluding a discrete set of frequencie
 s\, $\\Gamma_1$ and $\\Gamma_2$ can again be differentiated from overdeter
 mined boundary data.\\\\
LOCATION:Seminar Room 1\, Newton Institute
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