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Homogenization of integral energies under periodically oscillating differential constraints

Abstract

A homogenization result for a family of integral energies

$$\begin{aligned} u_{\varepsilon }\mapsto \int _\Omega f(u_{\varepsilon }(x))\,dx,\quad \varepsilon \rightarrow 0^+, \end{aligned}$$

is presented, where the fields \(u_{\varepsilon }\) are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of \(\mathscr {A}\)-quasiconvexity with variable coefficients and on two-scale convergence techniques.

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Acknowledgments

The authors thank the Center for Nonlinear Analysis (NSF Grant No. DMS-0635983), where this research was carried out. The research of E. Davoli, and I. Fonseca was funded by the National Science Foundation under Grant No. DMS- 0905778. I. Fonseca was also supported by the National Science Foundation under Grant No. DMS-1411646. E. Davoli and I. Fonseca acknowledge support of the National Science Foundation under the PIRE Grant No. OISE-0967140. E. Davoli is a member of the INDAM-GNAMPA Project 2015 “Critical phenomena in the mechanics of materials: a variational approach”, and was supported by the Austrian FWF project “Global variational methods for nonlinear evolution equations”.

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Correspondence to Irene Fonseca.

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Communicated by L. Ambrosio.

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Davoli, E., Fonseca, I. Homogenization of integral energies under periodically oscillating differential constraints. Calc. Var. 55, 69 (2016). https://doi.org/10.1007/s00526-016-0988-5

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Mathematics Subject Classification

  • 49J45
  • 35D99
  • 49K20