A homogenization result for planar, polygonal networks
An inverse problem for the equation
Let be a bounded, convex planar domain whose boundary has a not too degenerate curvature. In this paper we provide partial answers to an identification question associated with the boundary value problem We prove two results: 1) If is not a ball and if one considers only solutions with , then there exist at most finitely many pairs of coefficients so that the normal derivative equals a given . 2) If one imposes no sign condition on the solutions but...
Homogenization limits of diffusion equations in thin domains
Regular inversion of the divergence operator with Dirichlet boundary conditions on a polygon
Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter
Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
We recently derived a very general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction (cf. Capdeboscq and Vogelius (2003)). In this paper we show how this representation formula may be used to obtain very accurate estimates for the size of the inhomogeneities in terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single...
A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction
We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.
A representation formula for the voltage perturbations caused by diametrically small conductivity inhomogeneities. Proof of uniform validity
Existence and blow up of solutions to certain classes of two-dimensional nonlinear Neumann problems
A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction
We establish an asymptotic representation formula for the steady state voltage perturbations caused by low volume fraction internal conductivity inhomogeneities. This formula generalizes and unifies earlier formulas derived for special geometries and distributions of inhomogeneities.
Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements
We recently derived a very general representation formula for the boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction ( Capdeboscq and Vogelius (2003)). In this paper we show how this representation formula may be used to obtain very accurate estimates for the size of the inhomogeneities in terms of multiple boundary measurements. As demonstrated by our computational experiments, these estimates are significantly better than previously known (single...
Asymptotic formulas for perturbations in the electromagnetic fields due to the presence of inhomogeneities of small diameter
We consider solutions to the time-harmonic Maxwell's Equations of a TE (transverse electric) nature. For such solutions we provide a rigorous derivation of the leading order boundary perturbations resulting from the presence of a finite number of interior inhomogeneities of small diameter. We expect that these formulas will form the basis for very effective computational identification algorithms, aimed at determining information about the inhomogeneities from electromagnetic boundary measurements. ...
Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume
In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.
Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume
In this paper we discuss the approximate reconstruction of inhomogeneities of small volume. The data used for the reconstruction consist of boundary integrals of the (observed) electromagnetic fields. The numerical algorithms discussed are based on highly accurate asymptotic formulae for the electromagnetic fields in the presence of small volume inhomogeneities.
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