Article

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

Communications in Partial Differential Equations (Impact Factor: 1.19). 09/2009; 35(11). DOI: 10.1080/03605300903564000
Source: arXiv

ABSTRACT In this paper, we study the perfect and the insulated conductivity problems with multiple inclusions imbedded in a bounded domain in $\mathbb{R}^n, n\ge 2$. 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.

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Available from: Yanyan Li, Jul 11, 2015
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    • "This paper is mainly concerned with the gradient estimate for a conductivity problem in three dimensions whose conductivity k degenerates to 0. The three dimensional insulating case has been regarded as a challenging problem. In [6], Bao-Li-Yin derived an upper bound of |∇u| with order 1 √ ǫ in three dimensions. To our best knowledge, there has not been any updated or improved result yet. "
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    ABSTRACT: In this paper, we consider a gradient estimate for a conductivity problem whose inclusions are two neighboring insulators in three dimensions. When inclusions with extreme conductivity (insulator or perfect conductor) are closely located, the gradients of solutions may become arbitrarily large in the narrow region in between inclusions as the distance between inclusions approaches zero. The estimate for gradient between insulators in three dimensions has been regarded as a challenging problem, while the optimal estimates in terms of the distance have been known for the other problems of perfectly conducting inclusions in two and higher dimensions, and insulators in two dimensions. In this paper, we establish an upper bound of gradient on the shortest line segment between two insulating unit spheres in three dimensions. It presents an improved dependency of gradient on the distance which is substantially different from the blow-up rates in the other extreme conductivity problems.
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    • "and corresponding bounds for the case of N > 2 particles and d > 3, see [6]. It is important to note that even though in some referred studies it was mentioned on what parameters the constant C in (1) depends upon, the precise asymptotics have not been captured, only bounds for it have been established. "
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    ABSTRACT: A heterogeneous medium of constituents with vastly different mechanical properties, whose inhomogeneities are in close proximity to each other, is considered. The gradient of the solution to the corresponding problem exhibits singular behavior (blow up) with respect to the distance between inhomogeneities. This paper introduces a concise procedure for capturing the leading term of gradient's asymptotics precisely. This procedure is based on a thorough study of the system's energy. The developed methodology allows for straightforward generalization to heterogeneous media with a nonlinear constitutive description.
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    • "In dimension d = 3 and d ≥ 4, the blow up rate of |∇u ∞ | turns out to be (ǫ| ln ǫ|) −1 and ǫ −1 respectively; see [11]. The results were extended to multi-inclusions in [12]. Further, more detailed, characterizations of the singular behavior of ∇u ∞ have been obtained by Ammari, Ciraolo, Kang, Lee and Yun [3], Ammari, Kang, Lee, Lim and Zribi [8], Bonnetier and Triki [13] [14], Kang, Lim and Yun [21] [22]. "
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    ABSTRACT: We establish upper bounds on the blow up rate of the gradients of solutions of the Lam\'e system with partially infinite coefficients in dimension two as the distance between the surfaces of discontinuity of the coefficients of the system tends to zero.
    Archive for Rational Mechanics and Analysis 01/2014; 215(1):307-351. DOI:10.1007/s00205-014-0779-0
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