Spectrum of the Neumann–Poincaré Operator and Optimal Estimates for Transmission Problems in the Presence of Two Circular Inclusions

Abstract We investigate the field concentration for conductivity equations in presence of closely located circular inclusions by exploiting spectral nature residing behind phenomenon concentration. This approach enables us not only to recover existing results with new insights but also produce significant results. known optimal estimates derivatives solution when conductivities have same relative signs, which show rates blow-up that if both is either 0 or $\infty $, then (including gradient) become arbitrarily large as distance between two tends 0. obtain different signs. Estimates obtained gradient bounded regardless inclusions, 2nd and higher may blow up zero one $0$ other $. examples are optimal.