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

Abstract

In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in R n , n ≥ 2. For these two extreme cases of the conductivity problems, the gradients of their solutions may blow up as two inclusions approach each other. We establish the gradient estimates for the perfect conductivity problems and an upper bound of the gradients for the insulated conductivity problems in terms of the distances between any two closely spaced inclusions. IntroductionIn 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. 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|)and ε −1 for n = 2, 3 and n ≥ 4 respectively. A criteria, in terms of a functional of boundary data, for the situation where blow-up rate is realized was also given. See e.g. the introductions of The partial differential equations for the conductivity problems arise also in the study of composite materials. In R 2 , as explained in [14], if we use the bounded domain to represent the cross-section of a fiber-reinforced composite and use the inclusions to represent the cross-sections of the embedded fibers, then by a standard anti-plane shear model, the conductivity equations can be derived, in which the electric potential corresponds to the out-of-plane elastic displacement and the electric field corresponds to the stress tensor. Therefore, the gradient estimates for the conductivity problems provide valuable information about the stress intensity inside the composite materials.When conductivities of the inclusions are away from zero and infinity, the boundedness of the gradients were observed numerically by Babuska, Anderson, Smith and Levin [4]. Bonnetier and Vogelius [6] proved it when the inclusions are two touching balls in R 2 . General results were established by Li and Vogelius [14] for second order divergence form elliptic equations with piecewise smooth coefficients,

Strong influence of a small fiber on shear stress in fiber-reinforced composites

“…In Bao, Li and Yin's paper [6], the blow-up rate has been also obtained in higher dimensional spaces. They have also done the follow-ups [4,7] that the blow-up rate known for a pair of fibers is also valid for any pair in the multiple inclusions in any dimensions. In their works [6,4,7], the inclusions are also of sufficiently general shapes as well as Yun's works [12,13].…”

“…It is worthy noting that the case of multiple inclusions has been investigated by Bao, Li and Yin [4,7]. Precisely speaking, they have considered each pair of adjacent inclusions in multiple inclusions.…”

“…Let D i , i = 1, 2, 3, be the balls defined as(4). Then (a) Optimal lower bound: There is a positive constant C independent of 1 , 2 , r 1 , r 2 = δ/2 and r 3 such that…”

Stress concentration for closely located inclusions in nonlinear perfect conductivity problems

“…(see for instance [6]). In the case of smooth inclusions, it has been proved that the optimal blow-up rate of |∇u| is δ −1/2 for N = 2, it is (δ| log δ|) −1 for N = 3 and δ −1 for N ≥ 4, see [1,2,3,5,6,7,8,9,10,21,22,23,24,28,29] and references therein. In addition to its mathematical interest, the characterization of the gradient blow-up is relevant for the applications in composite materials.…”

Interactions between moderately close circular inclusions: The Dirichlet–Laplace equation in the plane

“…Having in mind functionals involving values taken by the solution both near and far from the inhomogeneities, we are interested in providing a complete asymptotic expansion valid in the H 1 sense. The closely related question of pointwise estimates for the gradient of the solution in the case where two adjacent conductivity inclusions are nearly touching was also intensively studied recently, see [2,5,6] for example.…”