Gradient Estimates for the Perfect Conductivity Problem

Abstract

Let Ω be a bounded open set in R n with C 2,α boundary, n ≥ 2, 0 < α < 1, D 1 and D 2 be two bounded strictly convex open subsets in Ω with C 2,α boundaries which are ε apart and far away from ∂Ω, i.e.where κ 0 , r 0 > 0 are universal constants independent of ε. We denoteGiven ϕ ∈ C 2 (∂Ω), consider the following scalar equation with Dirichlet boundary condition:

Gradient Estimates for the Perfect and Insulated Conductivity Problems with Multiple Inclusions

“…For the derivation of the above equation, readers can refer to the Appendix of [5]. Note that the proof there is for k 1 = k 2 = · · · = k m , but it works also for the general case with modification.…”

“…Similar to the two inclusions case in [5], we first investigate the properties of v i (i = 0, 1, · · · , m), the matrix A = (a ij ) and the vector b defined by (2.11) and (2.12). Here we state the following lemma, for its proof, readers may refer to Lemma 2.4 in [5].…”

“…In this paper, a continuation of [5], we establish gradient estimates for the perfect conductivity problems in the presence of multiple closely spaced inclusions in a bounded domain in R n (n ≥ 2). We also establish an upper bound of the gradients for the insulated conductivity problems.…”

“…We also establish an upper bound of the gradients for the insulated conductivity problems. For these two extreme cases of the conductivity problems, the electric field, which is represented by the gradient of the solutions, may blow up as the inclusions approach to each other, the blow-up rates of the electric field have been studied in [1,3,5,19,20]. In particular, when there are only two strictly convex inclusions, and let ε be the distance between the two inclusions, then for the perfect conductivity problem, the optimal blow-up rates for the gradients, as ε approaches to zero, were established to be ε −1/2 , (ε| ln ε|)…”

“…See e.g. the introductions of [5] and [20] for a more detailed description of these results. More recently, Lim and Yun in [15] have obtained further estimates with explicit dependence of the blow-up rates on the size of the inclusions for the perfect conductivity problem (see also [1] for results of this type), and H. Ammari, H. Kang, H. Lee, M. Lim and H. Zribi in [2] have given more refined estimates of the gradient of solutions.…”