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Spherical mean transform: A PDE approach

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  • We study the spherical mean transform on $\mathbb{R}^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of $1+1$-dimension hyperbolic equations. Using these equations, we discuss two known problems. The first one is a local uniqueness problem investigated by M. Agranovsky and P. Kuchment, [ Memoirs on Differential Equations and Mathematical Physics, 52 (2011), 1--16]. We present a proof which only involves simple energy arguments. The second problem is to characterize the kernel of spherical mean transform on annular regions, which was studied by C. Epstein and B. Kleiner [ Comm. Pure Appl. Math., 46(3) (1993), 441--451]. We present a short proof that simultaneously provides the necessity and sufficiency for the characterization. As a consequence, we derive a reconstruction procedure for the transform with additional interior (or exterior) information.
        We also discuss how the approach works for the hyperbolic and spherical spaces.
    Mathematics Subject Classification: Primary: 53C65; Secondary: 35L15.


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