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Journal of Convex Analysis 28 (2021), No. 2, 613--628
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

For an energy integral of the calculus of variations we can read anisotropies and inhomogeneities looking at the analytic expression of the integrand. The non-homogeneous case give rise to homogenization, also well known under the names Gamma-convergence, G-convergence, H-convergence; we give some details, as well as we describe some connections between these notions and the Mosco-convergence. We also describe the connection of the Gamma-convergence 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: Gamma-convergence, Mosco-convergence, convergence of eigenvalues and eigenfunctions, elliptic equations, calculus of variations, regularity of solutions, p,q-growth conditions, general growth conditions.

MSC: 35B45, 35B65, 35J20, 35J60, 49J45, 49N60, 49R05.

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