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

• Published in: International mathematics research notices 2023-05, Vol.2023 (9), p.7638-7685
• Main Authors: Ji, Yong-Gwan, Kang, Hyeonbae
• Format: Article
• Language: English
• ISSN: 1073-7928; 1687-0247
• DOI: 10.1093/imrn/rnac057

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