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We consider the field concentration for the transmission problems of thehomogeneous and inhomogeneous conductivity equations in the presence of closelylocated circular inclusions. We revisit these well-studied problems byexploiting the spectral nature residing behind the phenomenon of the fieldconcentration. The spectral approach enables us not only to recover theexisting results with new insights but also to produce significant new results.We show that when relative conductivities of inclusions have different signs,then the gradient of the solution is bounded regardless of the distance betweeninclusions, but the second and higher derivatives may blow up as the distancetends to zero if one of conductivities is $0$ and the other $\infty$. Thisresult holds for both homogenous and inhomogeneous problems. We prove theseresults by precise quantitative estimates of the derivatives of the solution.We also show by examples that the estimates are optimal.

Spectrum of the Neumann-Poincar\'e operator and optimal estimates for transmission problems in presence of two circular inclusions