A-Quasiconvexity: Relaxation and Homogenization

Andrea Braides; Irene Fonseca; Giovanni Leoni

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Integral representation and Γ-convergence of variational integrals with -growth

Alessandra Coscia, Domenico Mucci (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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We study the integral representation properties of limits of sequences of integral functionals like $\int f (x, D u) d x$ under nonstandard growth conditions of -type: namely, we assume that ${| z |}^{p (x)} \leq f (x, z) \leq {L (1 + | z |}^{p (x)}) .$ Under weak assumptions on the continuous function , we prove -convergence to integral functionals of the same type. We also analyse the case of integrands depending explicitly on ; finally we weaken the assumption allowing to be discontinuous on nice sets.

Lower semicontinuity of multiple µ-quasiconvex integrals

Ilaria Fragalà (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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Lower semicontinuity results are obtained for multiple integrals of the kind $\int_{ℝ^{n}} f (x, \nabla_{μ} u) d μ$ , where is a given positive measure on $ℝ^{n}$ , and the vector-valued function belongs to the Sobolev space $H_{μ}^{1, p} (ℝ^{n}, ℝ^{m})$ associated with . The proofs are essentially based on blow-up techniques, and a significant role is played therein by the concepts of tangent space and of tangent measures to . More precisely, for fully general , a notion of quasiconvexity for along the tangent bundle to , turns out to be necessary for...