Evolutionary problems in non-reflexive spaces

Martin Kružík; Johannes Zimmer

Displaying similar documents to “Evolutionary problems in non-reflexive spaces”

DiPerna-Majda measures and uniform integrability

Martin Kružík (1998)

Commentationes Mathematicae Universitatis Carolinae

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The purpose of this note is to discuss the relationship among Rosenthal's modulus of uniform integrability, Young measures and DiPerna-Majda measures. In particular, we give an explicit characterization of this modulus and state a criterion of the uniform integrability in terms of these measures. Further, we show applications to Fatou's lemma.

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...

Nonconcentrating generalized Young functionals

Tomáš Roubíček (1997)

Commentationes Mathematicae Universitatis Carolinae

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The Young measures, used widely for relaxation of various optimization problems, can be naturally understood as certain functionals on suitable space of integrands, which allows readily various generalizations. The paper is focused on such functionals which can be attained by sequences whose “energy” (= $p$ th power) does not concentrate in the sense that it is relatively weakly compact in $L^{1} (Ω)$ . Straightforward applications to coercive optimization problems are briefly outlined.

Existence theorem for nonlinear micropolar elasticity

Josip Tambača, Igor Velčić (2010)

ESAIM: Control, Optimisation and Calculus of Variations

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In this paper we give an existence theorem for the equilibrium problem for nonlinear micropolar elastic body. We consider the problem in its minimization formulation and apply the direct methods of the calculus of variations. As the main step towards the existence theorem, under some conditions, we prove the equivalence of the sequential weak lower semicontinuity of the total energy and the quasiconvexity, in some variables, of the stored energy function.

On the measures of DiPerna and Majda

Martin Kružík, Tomáš Roubíček (1997)

Mathematica Bohemica

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DiPerna and Majda generalized Young measures so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in $L^{p}$ -spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally...