Abstract
We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation. For such a problem, that is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [ 4 ]. Starting from this we derive Lipschitz continuous dependence estimates for the corresponding inverse problem.
Highlights
Let Ω ⊂ Rd, d = 2, 3 be a bounded, connected, convex C1 domain and let m ωε = (1.1)i=1 where Bi = riB, i = 1, . . . , m, B is a given bounded smooth domain containing the origin and the inhomogeneities, i = 1, . . . , m are disjoint
We consider an inverse problem regarding the detection of small conductivity inhomogeneities in a boundary value problem for a semilinear elliptic equation
That is related to cardiac electrophysiology, an asymptotic expansion for the boundary potential due to the presence of small conductivity inhomogeneities was established in [4]
Summary
M, B is a given bounded smooth domain containing the origin and the inhomogeneities (zi + εriB), i = 1, . In [4] the following asymptotic expansion for (uε − u)|∂Ω has been derived m (uε − u)(y) = εd rid Mi(K0 − Ki)∇u(zi) · ∇xNu(zi, y) + u3(zi)Nu(zi, y) + o(εd), i=1. M; for some specific shapes, the polarization tensors can be explicitly computed (see, e.g., [2] for a detailed derivation). In order to detect the set of inhomogeneities in [5] the authors implemented a successful reconstruction algorithm based on the computation of the topological gradient of a suitable boundary misfit functional. The obtained result proves well-posedness of the inverse problem justifying mathematically the successful reconstructions obtained in [5].
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