On the relaxation of nonconvex superficial integral functionals
Copyright policy )On the relaxation of nonconvex superficial integral functionals
We present a new approach to the variational relaxation of functionals $F:D(\mathbb{R}^N;\mathbb{R}^m)\to[0,\infty[$ of the type:$$F(v):=\int_{\mathbb{R}^N}W(\nabla v(x))d\mu(x),$$where $W:\mathbb{R}^{mN}\to[0,\infty[$ is a continuous function with growth conditions of order $p\geq 1$ but not necessarily convex. We essentially study the case when $\mu$ is the $k$-dimensional Hausdorff measure restricted to a suitable piece of a $k$-dimensional smooth submanifold of $\mathbb{R}^N$.
Relaxation, Mathematics(all), Methods involving semicontinuity and convergence; relaxation, Applied Mathematics, low dimensional structures, [MATH] Mathematics [math], Quasiconvexity, Structure de faible dimension, quasiconvexity, integral functionals, relaxation, Intégrale par rapport à une mesure de Hausdorff, Variational problems in a geometric measure-theoretic setting, Low dimensional structure, Quasiconvexité, Existence theories for free problems in two or more independent variables, Integral with respect to an Hausdorff measure
Relaxation, Mathematics(all), Methods involving semicontinuity and convergence; relaxation, Applied Mathematics, low dimensional structures, [MATH] Mathematics [math], Quasiconvexity, Structure de faible dimension, quasiconvexity, integral functionals, relaxation, Intégrale par rapport à une mesure de Hausdorff, Variational problems in a geometric measure-theoretic setting, Low dimensional structure, Quasiconvexité, Existence theories for free problems in two or more independent variables, Integral with respect to an Hausdorff measure
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