In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance S between some inclusions tends to 0 and as the conductivity ...In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance S between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincare operator converge to ±1/2 as δ→ 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance 5 〉 0 from each other. When δ=0, the contact between the inclusions is of order m 〉 2. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann- Poincare operator, in terms of 5 and rn, and we check that we recover the estimates obtained in [10].展开更多
文摘In a composite medium that contains close-to-touching inclusions, the pointwise values of the gradient of the voltage potential may blow up as the distance S between some inclusions tends to 0 and as the conductivity contrast degenerates. In a recent paper [9], we showed that the blow-up rate of the gradient is related to how the eigenvalues of the associated Neumann-Poincare operator converge to ±1/2 as δ→ 0, and on the regularity of the contact. Here, we consider two connected 2-D inclusions, at a distance 5 〉 0 from each other. When δ=0, the contact between the inclusions is of order m 〉 2. We numerically determine the asymptotic behavior of the first eigenvalue of the Neumann- Poincare operator, in terms of 5 and rn, and we check that we recover the estimates obtained in [10].
