Lipschitz estimates and existence of correctors for nonlinearly elastic, periodic composites subject to small strains

  • Stefan Neukamm
  • Mathias SchäffnerEmail author


We consider periodic homogenization of nonlinearly elastic composite materials. Under suitable assumptions on the stored energy function (frame indifference; minimality, non-degeneracy and smoothness at identity; \(p\ge d\)-growth from below), and on the microgeometry of the composite (covering the case of smooth, periodically distributed inclusions with touching boundaries), we prove that in an open neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula that can be represented with help of a corrector. This generalizes a recent result of the authors by significantly relaxing the spatial regularity assumptions on the stored energy function. As an application, we consider the nonlinear elasticity problem for \(\varepsilon \)-periodic composites, and prove that minimizers (subject to small loading and well-prepared boundary data) satisfy a Lipschitz estimate that is uniform in \(0<\varepsilon \ll 1\). A key ingredient of our analysis is a new Lipschitz estimate (under a smallness condition) for monotone systems with spatially piecewise-constant coefficients. The estimate only depends on the geometry of the coefficient’s discontinuity-interfaces, but not on the distance between these interfaces.

Mathematics Subject Classification

35B27 74B20 35B65 



The authors acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project No. 405009441, and in the context of TU Dresden’s Institutional Strategy “The Synergetic University”.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Faculty of MathematicsTechnische Universität DresdenDresdenGermany
  2. 2.Institute of MathematicsLeipzig UniversityLeipzigGermany

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