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In this paper, we are concerned with the study of parabolic variational inequality. Under appropriate assumptions on the main functions, we obtain the existence of weak solutions after the construction of the penalized Young measure by Galerkin's method and the penalty method. The passage to the limit follows relying on the theory of Young measures. [ABSTRACT FROM AUTHOR]

We examine linear, time invariant, singularly perturbed differential equations, where a split into slow and fast variables is not prescribed. Simple linear algebra considerations give rise to a useful order reduction type framework. A comparison with the classical order reduction method is provided, and the relation to efficient computations is pointed out. [ABSTRACT FROM AUTHOR]

An existence result for quasilinear elliptic systems with nonstandard growth/coercive conditions, W - 1 L M ¯ -data and no sign condition is proved. [ABSTRACT FROM AUTHOR]

In this work, the existence result to a p-Kirchhoff-type parabolic system is considered. Based on Galerkin's method and the theory of Young measures, we prove the existence of weak solutions. [ABSTRACT FROM AUTHOR]

This work is devoted to studying the quasilinear elliptic system - d i v a (x , u , D u) + | u | p - 2 u + b (x , u , D u) = v (x) + f (x , u) + d i v g (x , u) on a bounded open domain of R n with homogeneous Dirichlet boundary conditions. We show that there is a weak solution to this system under regularity, growth, and coercivity conditions for a, but only with very moderate monotonicity assumptions. We prove the existence result by using Galerkin's approximation and the theory of Young measures. [ABSTRACT FROM AUTHOR]

We study a class of quasilinear elliptic systems in Sobolev spaces. We prove the existence of a weak solution via Young measures. [ABSTRACT FROM AUTHOR]

In this paper, we study the solvability of the initial-boundary value problem for quasilinear parabolic system in divergence form with nonstandard growth conditions related to N-functions. By means of the theory of Young measure we prove the existence of weak solutions. [ABSTRACT FROM AUTHOR]

In this paper, we will study the existence of weak solutions for a class of nonlinear p(x)-Laplace system. The techniques of Young measure valued solutions are used to achieve the needed results. [ABSTRACT FROM AUTHOR]

We prove the existence of solutions for a quasilinear elliptic system - div σ (x , u , D u) = f (x , u , D u) in Ω , u = 0 on ∂ Ω. The results are obtained in Orlicz–Sobolev spaces by means of the Young measures. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to study a class of boundary value problem governed by a fractional differential inclusion in a separable Banach space E D α u (t) + λ D α − 1 u (t) ∈ F (t , u (t) , D α − 1 u (t)) , t ∈ [ 0 , 1 ] I 0 + β u (t) t = 0 = 0 , u (1) = I 0 + γ u (1) $$\begin{array}{} \displaystyle \left\{ \begin{array}{lll} D ^\alpha u(t) +\lambda D^{\alpha-1 }u(t) \in F(t, u(t), D ^{\alpha-1}u(t)), \hskip 2pt t \in [0, 1] \\ I_{0^+}^{\beta }u(t)\left\vert _{t=0}\right. = 0, \quad u(1)=I_{0^+}^{\gamma }u(1) \end{array} \right. \end{array}$$ in both Bochner and Pettis settings, where α ∈ ]1, 2], β ∈ [0, 2 – α], λ ≥ 0, γ > 0 are given constants, Dα is the standard Riemann-Liouville fractional derivative, and F : [0, 1] × E × E → 2E is a closed valued multifunction. Topological properties of the solution set are presented. Applications to control problems and subdifferential operators are provided. [ABSTRACT FROM AUTHOR]

Since the boundary of every physical domain is covered by microscopic asperities, we replace such a domain by an approximating sequence of domains having rough boundaries. We consider the instationary Boussinesq equations in these approximating domains where the temperature satisfies certain Robin boundary conditions. In the limit we will observe an additional weight factor in the Robin boundary condition which reflects the rugosity of the boundaries. Motivated by this observation we deal with the converse situation. Indeed, we will show that every Robin boundary condition with a fixed weight factor can be generated in the asymptotic limit by a sequence of domains with rough boundaries. [ABSTRACT FROM AUTHOR]

A continuum evolutionary model for micromagnetics is presented that, beside the standard magnetic balance laws, includes thermomagnetic coupling. To allow conceptually efficient computer implementation, inspired by relaxation method of static minimization problems, our model is mesoscopic in the sense that possible fine spatial oscillations of the magnetization are modeled by means of Young measures. Existence of weak solutions is proved by backward Euler time discretization. [ABSTRACT FROM AUTHOR]

We study mechanical problems with multiple solutions and introduce a thermodynamic framework to formulate two different selection criteria in terms of macroscopic energy productions and fluxes. Simple examples for lattice motion are then studied to compare the implications for both resting and moving inhomogeneities. [ABSTRACT FROM AUTHOR]

Various models of microstructure in deformation gradient and its evolution arising in martensitic mechanically-induced isothermal phase transformation are surveyed and scrutinized, focusing on over-bridging of various scales of the problem and its numerical approximation. In particular, numerically efficient model of a relaxed problem is shown to be approximated by conventional but computationally less efficient model based on standard partial differential inequalities. [ABSTRACT FROM AUTHOR]

The theory of thin wires developed in Dret and Meunier (Comptes Rendus de l'Académie des Sciences. Série I. Mathématique 337:143-147, ) is adapted to phase-transforming materials with large elastic moduli in the sense discussed in James and Rizzoni (J Elast 59:399-436, ). The result is a one-dimensional constitutive model for shape memory wires, characterized by a small number of material constants. The model is used to analyze self-accommodated and detwinned microstructures and to study superelasticity. It also turns out that the model successfully reproduces the behavior of shape memory wires in experiments of restrained recovery (Tsoi et al. in Mater Sci Eng A 368:299-310, ; Tsoi in 50:3535-3544, ; S̆ittner et al. in Mater Sci Eng A 286:298-311, ; vokoun in Smart Mater Struct 12:680-685, ; Zheng and Cui in Intermetallics 12:1305-1309, ; Zheng et al. in J Mater Sci Technol 20(4):390-394, ). In particular, the model is able to predict the shift to higher transformation temperatures on heating. The model also captures the effect of prestraining on the evolution of the recovery stress and of the martensite volume fraction. [ABSTRACT FROM AUTHOR]

Concerning a bounded sequence of finite energy weak solutions to the compressible Navier-Stokes-Poisson system (denoted by CNSP), which converges up to extraction of a subsequence, the limit system may not be the same system. By introducing Young measures as in [6, 15], the authors deduce the system (HCNSP) which the limit functions must satisfy. Then they solve this system in a subclass where Young measures are convex combinations of Dirac measures, to give the information on the propagation of density-oscillations. The results for strong solutions to (CNSP) (see Corollary 6.1) are also obtained. [ABSTRACT FROM AUTHOR]

It is pointed out that Corollary 1 in a recent paper by Khan et al. (Int J Game Theory 34:91–104, 2006), presented there as an extension of the Dvoretzky–Wald–Wolfowitz theorem, is a special case of Lyapunov’s theorem for Young measures (Balder in Rend Instit Mat Univ Trieste 31 Suppl. 1:1–69) It is also pointed out that Theorems 1–4 in Khan et al. (Int J Game Theory 34:91–104, 2006) follow from a single strong purification per se result that is already contained, as an implementation of that Lyapunov theorem for Young measures, in the proof of Theorem 2.2.1 in Balder (J Econ Theory 102:437–470, 2002). [ABSTRACT FROM AUTHOR]

This work is devoted to the study of two-scale gradient Young measures naturally arising in nonlinear elasticity homogenization problems. Precisely, a characterization of this class of measures is derived and an integral representation formula for homogenized energies, whose integrands satisfy very weak regularity assumptions, is obtained in terms of two-scale gradient Young measures. [ABSTRACT FROM AUTHOR]

We prove theorems on the lower semicontinuity and integral representations of the lower semicontinuous envelopes of integral functionals with integrandsLof fast growth:c1G(|Du|) +c2 =L=c3G(|Du|) +c4 withc3 =c1>0 andG: [0, 8[? [0, 8[ is an increasing convex function such thatvG' (v)/G(v)? 8 asv? 8 and is increasing for largev. Repeating the results for the case of the standard growth (G(·) = |·|p) the quasiconvexity of integrands characterizes the lower semicontinuity of integral functionals and their quasiconvexifications yield the integral functionals that are lower semicontinuous envelopes. [ABSTRACT FROM AUTHOR]

An optimal design problem for a plate governed by a linear, elliptic equation with bounded thickness varying only in a single prescribed direction and with unilateral isoperimetrical-type constraints is considered. Using Murat-Tartar’s homogenization theory for stratified plates and Young-measure relaxation theory, smoothness of the extended cost and constraint functionals is proved, and then the maximum principle necessary for an optimal relaxed design is derived. [ABSTRACT FROM AUTHOR]

The passage from atomistic to continuum models is usually done via $\Gamma$-convergence with respect to the weak topology of some Sobolev space; the obtained continuum energy, in a one dimensional model, is then convex. These kind of results are not optimal for problems related to materials which may undergo to phase transitions. We present here a new simple way for dealing with these problems. Our method consists in rewriting the discrete energy in terms of particular measures and taking the $\Gamma$-limit with respect to the weak * convergence of measures. The continuum energy arising from a linear chain of discrete mass points interacting with only the nearest neighbours turns out to be written in terms of Young measures. While, if the discrete mass points interact not only with the nearest neighbours but also with the second nearest neighbours we obtain a continuum problem in which appears a ``multiple Young measure" representing multiple levels of interaction. In this way we obtain a novel continuum problem which is able to capture the ``microstructure" at two different levels. [ABSTRACT FROM AUTHOR]

We deal with variational problems on varying manifolds in ℝn. We represent each manifold by a positive measure μ, to which we associate a suitable notion of tangent space Tμ, of mean curvature H(μ), and of Sobolev spaces with respect to μ on an open subset Ω ⊆ ℝn. We introduce the notions of weak and strong convergence for functions defined on varying manifolds, that is defined μh -a.e., being {μh} a weakly convergent sequence of measures. In this setting, we prove a strong-weak type compactness theorem for the pairs (Pμ h H(μh)), where Pμ h are the projectors onto the tangent spaces Tμ h. When μh belong to a suitable class of k-dimensional measures, having in particular a prescribed (k−1)-manifold as a boundary, we enforce this result to study the convergence of energy functionals, possibly with a Dirichlet condition on ∂Ω. We also address a perspective for optimization problems where the control variable is represented by a manifold with a prescribed boundary. [ABSTRACT FROM AUTHOR]

The paper establishes the continuity of the best approximation, or the projection, of a function in L for p∈[1,∞), on a closed convex set in the space, when the set varies and converges to a limit set in the Young-measure relaxation of the space. To this end a strong-type convergence and a convexity structure are identified on the space of Young measures. The appropriate convergence of sets with respect to which the continuity holds is the Mosco convergence of sets associated with the strong-type convergence of functions. [ABSTRACT FROM AUTHOR]

Generalizing techniques developed by Cuesta and Matrán for Bochner integrable random vectors of a separable Banach space, we prove a strong law of large numbers for Pettis integrable random elements of a separable locally convex space E. This result may be seen as a compactness result in a suitable topology on the set of Pettis integrable probabilities on E. [ABSTRACT FROM AUTHOR]

The asymptotic expansions are studied for the vorticity $$\{ \omega ^ \in (t,x)\} $$ to 2D incompressible Euler equations with-initial vorticity $$\omega _0^ \in (x) = \omega _0 (x) + \varepsilon \omega _0^1 \left( {x,\frac{{\varphi _0 (x)}}{\varepsilon }} \right)$$ , where ϕ0( x) satisfies |d ϕ0( x)|≠0 on the support of $$\omega _0^1 \left( { \cdot ,\theta } \right),\theta \in {\rm T}$$ and $$\omega _0 \left( x \right)(resp. \omega _0^1 (x,\theta ))$$ is sufficiently smooth and with compact support in ℝ2 (resp. ℝ2× T) The limit, v( t, x), of the corresponding velocity fields { v ɛ( t,x)} is obtained, which is the unique solution of (E) with initial vorticity ω0( x). Moreover, $$\omega ^ \in (t,x) = \omega (t,x) + \varepsilon \omega ^1 \left( {t,x,\frac{{\varphi (t,x)}}{\varepsilon }} \right) + o(\varepsilon ){\text{ }}in{\text{ }}C([0,\infty ),{\text{ L}}^p $$ (ℤ2)) for all 1≽ p∞, where $$\omega (t,x) = \partial _1 \upsilon _2 (t,x) - \partial _2 \upsilon _1 \left( {t,x} \right),\omega ^1 (t,x,\theta )$$ and ϕ( t,x) satisfy some modulation equation and eikonal equation, respectively. [ABSTRACT FROM AUTHOR]

Young-measure type limits of multifunctions are examined. The relaxed limit via selections is a multimeasure, while the relaxation as a set-valued point function yields a Young measure on sets. The paper verifies that the two processes are equivalent in the sense that the limit multimeasure consists of all the distributions selectionable with respect to the Young measure on sets. The main tool is the characterization of selectionable distributions. An illustrative application of the equivalence result is provided. [ABSTRACT FROM AUTHOR]

A well-known equivalence of randomization result of Wald and Wolfowitz states that any Young measure can be regarded as a probability measure on the set of all measurable functions. Here we give a sufficient condition for the Young measure to be equivalent to a probability measure on the set of all integrable selectors of a given multifunction. In this way, Aumann's identity for integrals of multifunctions can be interpreted in a novel fashion. By additionally applying a fundamental result from Young measure theory to uniformly L-bounded sequences of functions, Fatou's lemma in several dimensions, which is formulated in terms of the integral of a Kuratowski limes superior multifunction, can be proven in a new fashion. Also, a natural extension of these arguments leads to a generalization of a recent result by Artstein and Rzezuchowski [3]. [ABSTRACT FROM AUTHOR]

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