Multiscale Relaxation of Convex Functional
Multiscale Relaxation of Convex Functional
Summary: The \(\Gamma\)-limit of a family of functionals \[ u\mapsto \int_\Omega f\Biggl({x\over\varepsilon}, {x\over \varepsilon^2}, D^su\Biggr)\,dx \] is obtained for \(s= 1,2\) and when the integrand \(f= f(x,y,v)\) is a continuous function, periodic in \(x\) and \(y\), and convex with respect to \(v\). The 3-scale limits of second-order derivatives are characterized.
measurability selection criterion, \({\mathcal A}\)-quasiconvexity, Gamma-convergence, convexity, Convexity, homogenization, two-scale convergence, Methods involving semicontinuity and convergence; relaxation, Set-valued set functions and measures; integration of set-valued functions; measurable selections, periodicity, \(\Gamma\)-convergence, multiscale limits
measurability selection criterion, \({\mathcal A}\)-quasiconvexity, Gamma-convergence, convexity, Convexity, homogenization, two-scale convergence, Methods involving semicontinuity and convergence; relaxation, Set-valued set functions and measures; integration of set-valued functions; measurable selections, periodicity, \(\Gamma\)-convergence, multiscale limits
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