Analyticity of Homogenized Coefficients Under Bernoulli Perturbations and the Clausius–Mossotti Formulas

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• 作者：Mitia Duerinckx ; Antoine Gloria
• 刊名：Archive for Rational Mechanics and Analysis
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：220
• 期：1
• 页码：297-361
• 全文大小：987 KB
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• 作者单位：Mitia Duerinckx (1)
  Antoine Gloria (1) (2)
  
  1. Département de Mathématique, Université Libre de Bruxelles, Brussels, Belgium
  2. MEPHYSTO Team, Inria Lille-Nord Europe, Villeneuve d’Ascq, France
  
• 刊物类别：Physics and Astronomy
• 刊物主题：Physics
  Mechanics
  Electromagnetism, Optics and Lasers
  Mathematical and Computational Physics
  Complexity
  Fluids
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1432-0673

文摘

This paper is concerned with the behavior of the homogenized coefficients associated with some random stationary ergodic medium under a Bernoulli perturbation. Introducing a new family of energy estimates that combine probability and physical spaces, we prove the analyticity of the perturbed homogenized coefficients with respect to the Bernoulli parameter. Our approach holds under the minimal assumptions of stationarity and ergodicity, both in the scalar and vector cases, and gives analytical formulas for each derivative that essentially coincide with the so-called cluster expansion used by physicists. In particular, the first term yields the celebrated (electric and elastic) Clausius–Mossotti formulas for isotropic spherical random inclusions in an isotropic reference medium. This work constitutes the first general proof of these formulas in the case of random inclusions. Communicated by C. Le Bris