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Displaying similar documents to “Nonconcentrating generalized Young functionals”

Optimality conditions for nonconvex variational problems relaxed in terms of Young measures

Tomáš Roubíček (1998)

Kybernetika

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The scalar nonconvex variational problems of the minimum-energy type on Sobolev spaces are studied. As the Euler–Lagrange equation dramatically looses selectivity when extended in terms of the Young measures, the correct optimality conditions are sought by means of the convex compactification theory. It turns out that these conditions basically combine one part from the Euler–Lagrange equation with one part from the Weierstrass condition.

Evolutionary problems in non-reflexive spaces

Martin Kružík, Johannes Zimmer (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

On convergence of gradient-dependent integrands

Martin Kružík (2007)

Applications of Mathematics

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We study convergence properties of { v ( ∇ u k ) } k ∈ ℕ if v ∈ C ( ℝ m × n ) , | v ( s ) | ≤ C ( 1 + | s | p ) , 1 < p < + ∞ , has a finite quasiconvex envelope, u k → u weakly in W 1 , p ( Ω ; ℝ m ) and for some g ∈ C ( Ω ) it holds that ∫ Ω g ( x ) v ( ∇ u k ( x ) ) d x → ∫ Ω g ( x ) Q v ( ∇ u ( x ) ) d x as k → ∞ . In particular, we give necessary and sufficient conditions for L 1 -weak convergence of { det ∇ u k } k ∈ ℕ to det ∇ u if m = n = p .

Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

Irene Fonseca, Martin Kružík (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps { u k } k ∈ ℕ ⊂ L p ( Ω ; ℝ m ) satisfying a linear differential constraint 𝒜 u k = 0 . Applications to sequential weak lower semicontinuity of integral functionals on 𝒜 -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det ∇ ϕ k ⇀ * det ∇ ϕ in measures on the closure of Ω ⊂ ℝ n if ϕ k ⇀ ϕ in W 1 , n ( Ω ; ℝ n ) . This convergence holds, for...