Embedded eigenvalues for the Neumann-Poincare operator

Eigenvalues of the Neumann-poincaré Operator for Two Inclusions

In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance δ between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincaré operator converge ...

Homogenization of the Eigenvalues of the Neumann-poincaré Operator

In this article, we investigate the spectrum of the Neumann-Poincaré operator associated to a periodic distribution of small inclusions with size ε, and its asymptotic behavior as the parameter ε vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the ‘trivial’ eigenvalues 0 and 1, and of a subset which ...

Contraction Properties of the Poincare Series Operator

Tau Functions for the Dirac Operator on the Poincare Disk

In this paper we define tau functions for holonomic fields associated with the Dirac operator on the Poincare disk. The deformation analysis of the tau functions is worked out and in the case of the two point function, the tau function is expressed in terms of a Painleve function of type VI.

Maximising Neumann eigenvalues on rectangles

We obtain results for the spectral optimisation of Neumann eigenvalues on rectangles in R with a measure or perimeter constraint. We show that the rectangle with measure 1 which maximises the k’th Neumann eigenvalue converges to the unit square in the Hausdorff metric as k → ∞. Furthermore, we determine the unique maximiser of the k’th Neumann eigenvalue on a rectangle with given perimeter. AMS...