
Journal of Convex Analysis 28 (2021), No. 2, 613628 Copyright Heldermann Verlag 2021 Anisotropy Versus Inhomogeneity in the Calculus of Variations Paolo Marcellini Dip. di Matematica e Informatica "U. Dini", University of Firenze, Italy paolo.marcellini@unifi.it For an energy integral of the calculus of variations we can read anisotropies and inhomogeneities looking at the analytic expression of the integrand. The nonhomogeneous case give rise to homogenization, also well known under the names Gammaconvergence, Gconvergence, Hconvergence; we give some details, as well as we describe some connections between these notions and the Moscoconvergence. We also describe the connection of the Gammaconvergence with the convergence of eigenvalues and eigenfunctions. The anisotropic case for an energy integral of the calculus of variations appears when the integrand has different behaviors in different directions of the space; we consider anisotropic energy integrals as well as integrals with more general growth. Keywords: Gammaconvergence, Moscoconvergence, convergence of eigenvalues and eigenfunctions, elliptic equations, calculus of variations, regularity of solutions, p,qgrowth conditions, general growth conditions. MSC: 35B45, 35B65, 35J20, 35J60, 49J45, 49N60, 49R05. [ Fulltextpdf (147 KB)] for subscribers only. 