Young measures

Valadier, Michel

doi:10.1007/bfb0084935

Cited by 107 publications

(66 citation statements)

References 33 publications

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“…[Bal84,Bal89,Val90] Preliminarily, we establish some basic notation and definitions: We denote by L (0,T ) the σ-algebra of the Lebesgue measurable subsets of the interval (0, T ) and, given a reflexive Banach space V , by B(V ) its Borel σ-algebra. We use the symbol ⊗ for product σ-algebrae.…”

Section: A Young Measure Toolsmentioning

confidence: 99%

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Knees

Rossi

Zanini

2015

Nonlinear Analysis: Real World Applications

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This paper focuses on rate-independent damage in elastic bodies. Since the driving energy is nonconvex, solutions may have jumps as a function of time, and in this situation it is known that the classical concept of energetic solutions for rate-independent systems may fail to accurately describe the behavior of the system at jumps.Therefore we resort to the (by now well-established) vanishing viscosity approach to rate-independent modeling, and approximate the model by its viscous regularization. In fact, the analysis of the latter PDE system presents remarkable difficulties, due to its highly nonlinear character. We tackle it by combining a variational approach to a class of abstract doubly nonlinear evolution equations, with careful regularity estimates tailored to this specific system, relying on a q-Laplacian type gradient regularization of the damage variable. Hence for the viscous problem we conclude the existence of weak solutions, satisfying a suitable energy-dissipation inequality that is the starting point for the vanishing viscosity analysis. The latter leads to the notion of (weak) parameterized solution to our rate-independent system, which encompasses the influence of viscosity in the description of the jump regime.

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Section: A Young Measure Toolsmentioning

confidence: 99%

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Knees

Rossi

Zanini

2015

Nonlinear Analysis: Real World Applications

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“…x ∈Ω the collection {ν x } x∈Ω is the so-called Young measure on (Ω, σ) ( [43], see also [5,36,39,41,42]). …”

Section: Definition and Basic Propertiesmentioning

confidence: 99%

Oscillations and concentrations in sequences of gradients

Kałamajska

Kružík

2007

ESAIM: COCV

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Abstract.We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, {∇u k }, bounded in L p (Ω; R m×n ) if p > 1 and Ω ⊂ 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 results by Meyers and by Acerbi and Fusco are also discussed.Mathematics Subject Classification. 49J45, 35B05.

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“…We define such controls via Young measures. A brief discussion of Young measures is given in Appendix; for a more solid introduction to the subject see, for instance, [22].…”

Section: Generalized Controlsmentioning

confidence: 99%

Optimal control of continuity equations

Pogodaev

2016

Nonlinear Differ. Equ. Appl.

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An optimal control problem for the continuity equation is considered. The aim of a "controller" is to maximize the total mass within a target set at a given time moment. The existence of optimal controls is established. For a particular case of the problem, where an initial distribution is absolutely continuous with smooth density and the target set has certain regularity properties, a necessary optimality condition is derived. It is shown that for the general problem one may construct a perturbed problem that satisfies all the assumptions of the necessary optimality condition, and any optimal control for the perturbed problem, is nearly optimal for the original one.2000 Mathematics Subject Classification: 49K20, 49J15

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Young measures

Cited by 107 publications

References 33 publications

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

A quasilinear differential inclusion for viscous and rate-independent damage systems in non-smooth domains

Oscillations and concentrations in sequences of gradients

Optimal control of continuity equations

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