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Homogenization of the Brush Problem with a Source Term in L 1

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Abstract

We consider a domain which has the form of a brush in 3D or the form of a comb in 2D, i.e. an open set which is composed of cylindrical vertical teeth distributed over a fixed basis. All the teeth have a similar fixed height; their cross sections can vary from one tooth to another and are not supposed to be smooth; moreover the teeth can be adjacent, i.e. they can share parts of their boundaries. The diameter of every tooth is supposed to be less than or equal to \({\varepsilon}\), and the asymptotic volume fraction of the teeth (as \({\varepsilon}\) tends to zero) is supposed to be bounded from below away from zero, but no periodicity is assumed on the distribution of the teeth. In this domain we study the asymptotic behavior (as \({\varepsilon}\) tends to zero) of the solution of a second order elliptic equation with a zeroth order term which is bounded from below away from zero, when the homogeneous Neumann boundary condition is satisfied on the whole of the boundary. First, we revisit the problem where the source term belongs to L 2. This is a classical problem, but our homogenization result takes place in a geometry which is more general that the ones which have been considered before. Moreover we prove a corrector result which is new. Then, we study the case where the source term belongs to L 1. Working in the framework of renormalized solutions and introducing a definition of renormalized solutions for degenerate elliptic equations where only the vertical derivative is involved (such a definition is new), we identify the limit problem and prove a corrector result.

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Authors and Affiliations

  1. DIEI, Università degli Studi di Cassino e del Lazio Meridionale, via G. Di Biasio 43, 03043, Cassino, FR, Italy

    Antonio Gaudiello

  2. Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS, Université de Rouen, Avenue de l’Université, B.P. 12, Technopôle du Madrillet, 76801, Saint-Étienne-du-Rouvray, France

    Olivier Guibé

  3. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie (UPMC Paris VI), Boîte courrier 187, 75252, Paris Cedex 05, France

    François Murat

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  1. Antonio Gaudiello
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Correspondence to Antonio Gaudiello.

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Communicated by D. Kinderlehrer

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Gaudiello, A., Guibé, O. & Murat, F. Homogenization of the Brush Problem with a Source Term in L 1 . Arch Rational Mech Anal 225, 1–64 (2017). https://doi.org/10.1007/s00205-017-1079-2

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