Gradient Estimates for the Perfect Conductivity Problem

By Ellen Shiting Bao, Yanyan Li, Biao Yin. 2006

163 citations recorded by the source corpus.

We establish both upper and lower bounds of the gradient estimates for the perfect conductivity problem in the case where perfect (stiff) conductors are closely spaced inside an open bounded domain and away from the boundary. These results give the optimal blow-up rates of the stress for conductors with arbitrary shape and in all dimensions.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/8f4182cd46a5b51ef881df138525cd48f02900df

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Gradient estimates for solutions of the Lam\'e system with partially infinite coefficients in dimensions greater than two (2016, background): “,∂Ω) >κ1 3 o with ϕ′ := u ∂Ω′ . Remark 1.4. Since the blow up rate of |∇u∞| for solutions of the scalar equation (1.1) when k = ∞ is known to reach the magnitude (ǫ|lnǫ|)−1 in dimension three, see [11], estimate (1.11) is expected to be optimal. Following arguments in the proof of Theorem 1.1, we”
On the Spectrum of the Poincaré Variational Problem for Two Close-to-Touching Inclusions in 2D (2013, methodology): “This result was obtained in [9] using a variational approach.”
Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity (2012, background): “As shown in [3,5,7,11,17,22,23], in two dimensions the generic rate of gradient blow-up is ε−1/2, while it is |ε log ε|−1 in three dimensions [7,8,20].”
Gradient Estimates for Solutions of the Lamé System with Partially Infinite Coefficients (2013, background): “1(eΩ) ∩C1(D 1 ∪ D2). The convergence (1.4) in the case µ1 → ∞ while λ1 remains bounded was established in [6]. Our proof of (1.4) in the Appendix is diﬀerent and is an extension to systems of that in [11]. The solution of (1.5) is also the unique function which has the least energy in appropriate fu”