Young measure techniques for existence of Cournot-Nash-Walras equilibria

1999 ◽  
pp. 31-39 ◽  
Author(s):  
Erik Balder
Keyword(s):  
Young Measure

scholarly journals Cohesive fracture with irreversibility: Quasistatic evolution for a model subject to fatigue

2018 ◽  
Vol 28 (07) ◽  
pp. 1371-1412 ◽  
Author(s):  
Vito Crismale ◽  
Giuliano Lazzaroni ◽  
Gianluca Orlando
Keyword(s):  
Crack Surface ◽  
Crack Opening ◽  
Young Measure ◽  
Small Strain ◽  
Weak Formulation ◽  
Cohesive Fracture ◽  
Quasistatic Evolution ◽  
Antiplane Elasticity ◽  
Energy Dissipated ◽  
Model Subject

In this paper we prove the existence of quasistatic evolutions for a cohesive fracture on a prescribed crack surface, in small-strain antiplane elasticity. The main feature of the model is that the density of the energy dissipated in the fracture process depends on the total variation of the amplitude of the jump. Thus, any change in the crack opening entails a loss of energy, until the crack is complete. In particular this implies a fatigue phenomenon, i.e. a complete fracture may be produced by oscillation of small jumps. The first step of the existence proof is the construction of approximate evolutions obtained by solving discrete-time incremental minimum problems. The main difficulty in the passage to the continuous-time limit is that we lack of controls on the variations of the jump of the approximate evolutions. Therefore we resort to a weak formulation where the variation of the jump is replaced by a Young measure. Eventually, after proving the existence in this weak formulation, we improve the result by showing that the Young measure is concentrated on a function and coincides with the variation of the jump of the displacement.

A necessary and sufficient condition for nonattainment and formation of microstructure almost everywhere in scalar variational problems

1994 ◽  
Vol 124 (3) ◽  
pp. 437-471 ◽  
Author(s):  
Gero Friesecke
Keyword(s):  
Solid Phase ◽  
Young Measure ◽  
Variational Problems ◽  
Boundary Values ◽  
Necessary And Sufficient Condition ◽  
Linear Deformation ◽  
Almost Everywhere ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Unique Limit

For scalar variational problemssubject to linear boundary values, we determine completely those integrandsW: ℝn→ ℝ for which the minimum is not attained, thereby completing previous efforts such as a recent nonexistence theorem of Chipot [9] and unifying a large number of examples and counterexamples in the literature.As a corollary, we show that in case of nonattainment (and providedWgrows superlinearly at infinity), every minimising sequence converges weakly but not strongly inW1,1(Ω) to a unique limit, namely the linear deformation prescribed at the boundary, and develops fine structure everywhere in Ω, that is to say every Young measure associated with the sequence of its gradients is almost-nowhere a Dirac mass.Connections with solid–solid phase transformations are indicated.

Singular Perturbation as a Selection Criterion for Young‐Measure Solutions

2007 ◽  
Vol 39 (1) ◽  
pp. 195-209 ◽  
Author(s):  
M. Lilli ◽  
T. J. Healey ◽  
H. Kielhöfer
Keyword(s):  
Singular Perturbation ◽  
Selection Criterion ◽  
Young Measure

scholarly journals Conservation laws driven by Lévy white noise

2015 ◽  
Vol 12 (03) ◽  
pp. 581-654 ◽  
Author(s):  
Imran H. Biswas ◽  
Kenneth H. Karlsen ◽  
Ananta K. Majee
Keyword(s):  
Conservation Laws ◽  
Entropy Solution ◽  
Young Measure ◽  
Entropy Solutions ◽  
Contraction Principle ◽  
Viscosity Method ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Lévy White Noise ◽  
Entropy Inequalities

We consider multidimensional conservation laws perturbed by multiplicative Lévy noise. We establish existence and uniqueness results for entropy solutions. The entropy inequalities are formally obtained by the Itó–Lévy chain rule. The multidimensionality requires a generalized interpretation of the entropy inequalities to accommodate Young measure-valued solutions. We first prove the existence of entropy solutions in the generalized sense via the vanishing viscosity method, and then establish the L1-contraction principle. Finally, the L1 contraction principle is used to argue that the generalized entropy solution is indeed the classical entropy solution.

scholarly journals Numerical approximation of young-measure solutions to parabolic systems of forward-backward type

2019 ◽  
Vol 13 (3) ◽  
pp. 649-696
Author(s):  
Miles Caddick ◽  
Endre Süli
Keyword(s):  
Approximate Solution ◽  
Lipschitz Domain ◽  
Numerical Approximation ◽  
Young Measure ◽  
Large Data ◽  
Parabolic Systems ◽  
Lipschitz Mapping ◽  
Nonlinear Parabolic Systems ◽  
Nonlinear Parabolic ◽  
Locally Lipschitz

This paper is concerned with the proof of existence and numerical approximation of large-data global-in-time Young measure solutions to initial-boundaryvalue problems for multidimensional nonlinear parabolic systems of forward-backward type of the form ?tu - div(a(Du))+ Bu = F, where B ? Rmxm, Bv?v ? 0 for all v ? Rm, F is an m-component vector-function defined on a bounded open Lipschitz domain ? ? Rn, and a is a locally Lipschitz mapping of the form a(A)= K(A)A, where K: Rmxn ? R. The function a may have unequal lower and upper growth rates; it is not assumed to be monotone, nor is it assumed to be the gradient of a potential. We construct a numerical method for the approximate solution of problems in this class, and we prove its convergence to a Young measure solution of the system.

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