Spectrum of the Neumann-Poincare Operator and Optimal Estimates for Transmission Problems in the Presence of Two Circular Inclusions (Apr, 10.1093/imrn/rnac057, 2022)

초록

This is a correction to the paper [2]. In Theorems 1.3 and 1.4 of the paper, it is proved that the solution u satisfies the following estimate if (k1 − 1)(k2 − 1) < 0: ||u||n,0 ≤ C(4|λ1λ2| − 1 + r∗√∊)−n+1, (1) where ∊ is the small distance between two circular inclusions. It shows that the gradient is bounded regardless of ∊, but n-th order derivatives may blow up if n ≥ 2. It is claimed in those theorems that the estimate (1) is optimal in the sense that there is a case such that the reverse inequality holds for n = 2. Examples for optimality are given in Section 6 of the paper. However, there are errors in Section 6 and examples are not valid. The inequality (6.8) in [2] is not correct, which is caused by the sign errors in previous equations. For example, in the second line in p. 46, a – sign is needed after the equality symbol and the + sign in front of the second summation symbol needs to be changed to –. So, even if errors in Section 6 do not affect results in earlier sections and the estimate (1) is still valid, the optimality claim is not valid for n ≥ 2. In fact, it is proved that ||u||n,0 is bounded regardless of ∊ even for n ≥ 2 in the recent preprint [1].