We consider a Markov control model with Borel state space, metric compact action space, and transitions assumed to have a density function with respect to some probability measure satisfying some continuity conditions. We study the optimization problem of maximizing the probability of visiting some subset of the state space infinitely often, and we show that there exists an optimal stationary Markov policy for this problem. We endow the set of stationary Markov policies and the family of strategic probability measures with adequate topologies (namely, the narrow topology for Young measures and the $ws^\infty$-topology, respectively) to obtain compactness and continuity properties, which allow us to obtain our main results.

Let n, $N \in {\open N}$ with $\Omega \subseteq {\open R}^n$ open. Given ${\rm H} \in C^2(\Omega \times {\open R}^N \times {\open R}^{Nn})$, we consider the functional 1

$${\rm E}_\infty (u,{\rm {\cal O}})\, : = \, \mathop {{\rm ess}\,\sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u,{\rm D}u),\quad u\in W_{{\rm loc}}^{1,\infty} (\Omega, {\open R}^N),\quad {\rm {\cal O}}{\Subset}\Omega.$$

$L^\infty $

2

$$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0,\hfill}\right.$$

$[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$

${\cal D}$

A corrector theory for the strong approximation of gradient fields inside periodiccomposites made from two materials with different power law behavior is provided. Eachmaterial component has a distinctly different exponent appearing in the constitutive lawrelating gradient to flux. The correctors are used to develop bounds on the localsingularity strength for gradient fields inside micro-structured media. The bounds aremulti-scale in nature and can be used to measure the amplification of applied macroscopicfields by the microstructure. The results in this paper are developed for materials havingpower law exponents strictly between  −1 and zero.

A new approach to irreversible quasistatic fracture growth is given, by means of Young measures.The study concerns a cohesive zone model with prescribed crack path, when the material gives different responses to loading and unloading phases.In the particular situation of constant unloading response,the result contained in [G. Dal Maso and C. Zanini,Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 253–279] is recovered.In this case, the convergence of the discrete time approximationsis improved.

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $\{u_k\}_{k\in{\mathbb N}} \subset L^p(\Omega;{\mathbb R}^m)$ satisfying a linear differential constraint ${\mathcal A}u_k=0$ . Applications to sequential weak lower semicontinuity of integral functionals on ${\mathcal A}$ -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det $\nabla\varphi_k\stackrel{*}{\rightharpoonup}{\rm det}\nabla\varphi$ in measures on the closure of $\Omega\subset{\mathbb R}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^{1,n}(\Omega;{\mathbb R}^n)$ . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemmaprecisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_{\Omega\setminus\Omega_j} v(u(x))\,{\rm d}x$ along $\{u_k\}\subset L^p(\Omega;{\mathbb R}^m)\cap{\rm ker}\ {\mathcal A}$ . Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.

Rate-independent evolution for material models with nonconvexelastic energies is studied without any spatial regularization ofthe inner variable; due to lack of convexity, the model is developedin the framework of Young measures. An existence result for thequasistatic evolution is obtained in terms of compatible systems ofYoung measures. We also show as this result can be equivalentlyreformulated with probabilistic language and leads to thedescription of the quasistatic evolution in terms of stochasticprocesses on a suitable probability space.

This paper addresses the Cauchy problem for thegradient flow equation in a Hilbert space  $\mathcal{H}$ \[\begin{cases}u'(t)+ \partial_{\ell}\phi(u(t))\ni f(t)&\text{{\it a.e.}\ in }(0,T),u(0)=u_0, \end{cases}\] where $\phi: \mathcal{H} \to (-\infty,+\infty]$ is a proper,lower semicontinuous functional which is not supposed to be a(smooth perturbation of a) convex functional and $\partial_{\ell}\phi$ is(a suitable limiting version of) its subdifferential.We will present some new existence results for the solutions of theequation by exploiting a variational approximationtechnique, featuring some ideas from the theory of Minimizing Movementsand of Young measures.
Our analysisis also motivated by some models describing phase transitionsphenomena, leading tosystems of evolutionary PDEs which have a common underlying gradient flow structure:in particular, we will focus onquasistationary models, which exhibithighly non convex Lyapunov functionals.

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.

We establish two new formulations of the membrane problem by working in the space of $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -Young measures and $W^{1,p}_{\Gamma_0}(\Omega,\mathbf R^3)$ -varifolds. The energyfunctional related to these formulations is obtained as a limit of the 3d formulation of the behavior of a thin layer for a suitable variational convergence associated with the narrow convergence of Young measures and with some weak convergence of varifolds. Theinterest of the first formulation is to encode the oscillation informations on the gradients minimizing sequences related to the classicalformulation. The second formulation moreover accounts for concentration effects.

We give necessary and sufficient conditions which characterize the Young measures associated to two oscillating sequences of functions, un on $\omega_1\times \omega_2$ and vn on $\omega_2$ satisfying the constraint $v_n(y)=\frac{1}{|\omega_1|} \int_{\omega_1} u_n (x, y) dx$. Our study is motivated by nonlinear effects induced by homogenization. Techniques based on equimeasurability and rearrangements are employed.