On the spectrum of the Neumann Poincaré operator in a composite medium containing close-to-touching inclusions

Eric Bonnetier (1) , Faouzi Triki (1)

1 EDP - Equations aux Dérivées Partielles

The Neumann-Poincaré operator naturally appears in the integral formulation of elliptic transmission problems. This operator has received some attention in recent years in the context of metamaterials, and plasmonics. We study this operator in composite media containing close-to touching inhomogeneities. In the narrow channels between the inclusions, the gradients of the solution to the associated PDE may become unbounded, as the inter-inclusion distance tends to 0 and as the contrast of material coefficients becomes large. We study how the blow up of the gradient can be infered from the spectral properties of the Nemann-Poincaré operator when the inclusions are discs in 2D. We also consider the case of inclusions with $C^{m}$ contact, $m \geq 2$.

Mots clés: Composite media; blow up; Poincaré-Neumann operator

Domaines: Equations aux dérivées partielles [math.AP]

Eric Bonnetier, Faouzi Triki. On the spectrum of the Neumann Poincaré operator in a composite medium containing close-to-touching inclusions. Applied Inverse Problems Conference, Jul 2013, Daejon, South Korea. ⟨hal-00987200⟩