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CATEGORIES:Isaac Newton Institute Seminar Series
SUMMARY:On the crack inverse problem for pressure waves in
half-space - Darko Volkov (Worcester Polytechnic
Institute)
DTSTART;TZID=Europe/London:20230203T090000
DTEND;TZID=Europe/London:20230203T094500
UID:TALK194569AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/194569
DESCRIPTION:Starting from the pressure wave equation in half-s
pace minus a crack with a zero Neumann condition o
n the top plane\, we introduce a related inverse p
roblem. That inverse problem consists of identifyi
ng the crack and the unknown forcing term on that
crack from overdetermined boundary data on a relat
ively open set of the top plane. This inverse prob
lem is not uniquely solvable unless some additiona
l assumption is made. However\, we show that we ca
n differentiate two cracks $\\Gamma_1$ and $\\Gamm
a_2$ 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 th
at case requires using an advanced unique continua
tion result obtained by Barcelo et al.\, 1988. In
particular\, this unique continuation result impli
es 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 d
emonstrate non-uniqueness for the crack inverse pr
oblem\, even if $\\Gamma_1$ and $\\Gamma_2$ are sm
ooth and "almost" flat. This requires using argume
nts borrowed from the analysis of elliptic PDEs on
domains with cusps to verify that certain solutio
ns can be extended on the outside of these domains
.\\\\\nFinally\, we show in the case where $\\RR^3
\\sm \\ov{\\Gamma_1\\cup \\Gamma_2}$ is not neces
sarily connected that after excluding a discrete s
et of frequencies\, $\\Gamma_1$ and $\\Gamma_2$ ca
n again be differentiated from overdetermined boun
dary data.\\\\
LOCATION:Seminar Room 1\, Newton Institute
CONTACT:
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