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• W2059861409 startingPage "1540" @default.
• W2059861409 abstract "We consider integral functionals of the type $F(u):=int_{Omega} f(x,u,Du) dx$ exhibiting a gap between the coercivity and the growth exponent $$L^{-1}|Du|^pleq f(x,u,Du)leq L(1+|Du|^q), 1 < p < q,1leq L < + infty,.$$ We give lower semicontinuity results and conditions ensuring that the relaxed functional $ol{F}$ is equal to ,$int_{Omega} Qf(x,u,Du) dx$, where $Qf$ denotes the usual quasi-convex envelope; our conditions are sharp. Indeed, we also provide counterexamples where such an integral representation fails, showing that energy concentrations appear in the relaxation procedure leading to a measure representation of $ol{F}$ with a nonzero singular part, which is explicitly computed. The main point in our analysis is that such relaxation results depend in a subtle way on the interaction between the ratio $q/p$ and the degree of regularity of the integrand f with respect to the variable x. Our results extend theorems for nonconvex integrals due to Fonseca and Malý and Kristensen; the energies we treat are related to strongly anisotropic settings." @default.
• W2059861409 created "2016-06-24" @default.
• W2059861409 creator A5017963438 @default.
• W2059861409 creator A5055486184 @default.
• W2059861409 date "2005-01-01" @default.
• W2059861409 modified "2023-10-18" @default.
• W2059861409 title "Integral Functionals and the Gap Problem: Sharp Bounds for Relaxation and Energy Concentration" @default.
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• W2059861409 doi "https://doi.org/10.1137/s0036141003424113" @default.
• W2059861409 hasPublicationYear "2005" @default.
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