DiPerna-Majda measures and uniform integrability

Martin Kružík

Displaying similar documents to “DiPerna-Majda measures and uniform integrability”

Oscillations and concentrations in sequences of gradients

Martin Kružík, Agnieszka Kałamajska (2008)

ESAIM: Control, Optimisation and Calculus of Variations

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We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, ${\nabla u_{k}}$ , bounded in $L^{p} (Ø; ℝ^{m \times n})$ if $p > 1$ and $Ω \subset ℝ^{n}$ is a bounded domain with the extension property in $W^{1, p}$ . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $Ω$ are required and links with lower semicontinuity...

A characterization of gradient Young-concentration measures generated by solutions of Dirichlet-type problems with large sources

Gisella Croce, Catherine Lacour, Gérard Michaille (2009)

ESAIM: Control, Optimisation and Calculus of Variations

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We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order $\frac{1}{\sqrt{ε}}$ concentrated on an $ε$ -neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

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.

A refinement of Ball's theorem on Young measures.

Hüngerbuhler, Norbert (1997)

The New York Journal of Mathematics [electronic only]

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Homogenization of periodic nonconvex integral functionals in terms of Young measures

Omar Anza Hafsa, Jean-Philippe Mandallena, Gérard Michaille (2006)

ESAIM: Control, Optimisation and Calculus of Variations

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Homogenization of periodic functionals, whose integrands possess possibly multi-well structure, is treated in terms of Young measures. More precisely, we characterize the $Γ$ -limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.

Admissible functions in two-scale convergence.

Valadier, Michel (1997)

Portugaliae Mathematica

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Extremal length of vector measures.

Aikawa, Hiroaki, Ohtsuka, Makoto (1999)

Annales Academiae Scientiarum Fennicae. Mathematica

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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 ${u_{k}}_{k \in ℕ} \subset L^{p} (Ω; ℝ^{m})$ 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 $\nabla ϕ_{k} \overset{*}{⇀} \det \nabla ϕ$ in measures on the closure of $Ω \subset ℝ^{n}$ if $ϕ_{k} ⇀ ϕ$ in $W^{1, n} (Ω; ℝ^{n})$ . This convergence holds, for...