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Periodic homogenization of a non-coercive class of functionals. (Spanish. English summary) Zbl 0944.49016
Summary: During the last decade several authors proved results concerning the \(\Gamma\)-convergence of a class of non-coercive functionals of the form \[ J_\varepsilon= \int_\Omega f\Biggl(x,{x\over\varepsilon}, u(x),\nabla u(x)\Biggr) dx\quad\text{as }\varepsilon\to 0, \] under certain conditions on \(f\). This work uses those \(\Gamma\)-limit existence results to construct a formula which gives that limit when \(f(x,.,u,z)\) is \(Y\)-periodic. First the case \(f({x\over\varepsilon}, \nabla u)\) is studied, for which the proof in the non-coercive case is essentially the same as for the coercive case. From the study of this case there follow formulae for the general case considered here. The paper ends proving that, under certain special conditions for \(f\), for all \(\Omega\subset \mathbb{R}^N\) open and bounded and for all \(u\in W^{1,p}(\Omega)\), it holds that \[ J(u)= \int_\Omega\widehat f(x,u(x),\nabla u(x)) dx= \Gamma(L^p(\Omega)) \lim_{\varepsilon\to 0} \int_\Omega f\Biggl(x,{x\over\varepsilon}, u(x),\nabla u(x)\Biggr) dx, \] where \[ \widehat f(x,u,z)= \inf\Biggl\{\sim \int_Y f(x,y,u,\nabla w(y)+ z) dy: w\in W^{1,p}_{per}(Y)\Biggr\}. \] The physical interpretation is that, known the microscopic structure of a physical medium described by \(J_\varepsilon(u)\), then \(J(u)\) describes the macroscopic properties of such medium.
49J45 Methods involving semicontinuity and convergence; relaxation
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
74Q05 Homogenization in equilibrium problems of solid mechanics
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