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In this paper we provide a representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction in a simplified {em monodomain model} describing the electric activity of the heart. We derive such a result in the case of a nonlinear problem. Our long-term goal is the solution of the inverse problem related to the detection of regions affected by heart ischemic disease, whose position and size are unknown. We model the presence of ischemic regions in the form of small inhomogeneities. This leads to the study of a boundary value problem for a semilinear elliptic equation. We first analyze the well-posedness of the problem establishing some key energy estimates. These allow us to derive rigorously an asymptotic formula of the boundary potential perturbation due to the presence of the inhomogeneities, following an approach similar to the one introduced by Capdeboscq and Vogelius in cite{capvoge} in the case of the linear conductivity equation. Finally, we propose some ideas of the reconstruction procedure that might be used to detect the inhomogeneities.
In this paper we deal with the problem of determining perfectly insulating regions (cavities) from boundary measurements in a nonlinear elliptic equation arising from cardiac electrophysiology. With minimal regularity assumptions on the cavities, we
A regular elliptic boundary-value problem over a bounded domain with a smooth boundary is studied. We prove that the operator of this problem is a Fredholm one in the two-sided refined scale of the functional Hilbert spaces and generates a complete c
Given a smooth domain $OmegasubsetRR^N$ such that $0 in partialOmega$ and given a nonnegative smooth function $zeta$ on $partialOmega$, we study the behavior near 0 of positive solutions of $-Delta u=u^q$ in $Omega$ such that $u = zeta$ on $partialOm
We give necessary and sufficient conditions for the solvability of some semilinear elliptic boundary value problems involving the Laplace operator with linear and nonlinear highest order boundary conditions involving the Laplace-Beltrami operator.
In this paper, we consider the pointwise boundary Lipschitz regularity of solutions for the semilinear elliptic equations in divergence form mainly under some weaker assumptions on nonhomogeneous term and the boundary. If the domain satisfies C^{1,te