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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. [ABSTRACT FROM AUTHOR]

The present article deals with the existence of weak solutions to a class of p(z)p\left(z)-Kirchhoff-type problems. To address these problems, we employ a variational approach in conjunction with the theory of variable exponent Sobolev spaces, while imposing suitable assumptions on the source term. Furthermore, we utilize the theory of Young measures.

In deterministic and stochastic control theory, relaxed or randomized control policies allow for versatile mathematical analysis (on continuity, compactness, convexity, and approximations) to be applicable with no artificial restrictions on the classes of control policies considered, leading to very general existence results on optimal measurable policies under various setups and information structures. On relaxed controls, two studied topologies are the Young and Borkar (weak\ast) topologies on spaces of functions from a state/measurement space to the space of probability measures on control action spaces; the former via a weak convergence topology on probability measures on a product space with a fixed marginal on the input (state) space, and the latter via a weak\ast topology on randomized policies viewed as maps from states/measurements to the space of signed measures with bounded variation. We establish implication and equivalence conditions between the Young and Borkar topologies on control policies. We then show that, under some conditions, for a controlled Markov chain with standard Borel spaces the invariant measure is weakly continuous on the space of stationary control policies defined by either of these topologies. An implication is near-optimality of quantized stationary policies in state and actions or continuous stationary and deterministic policies for average cost control under two sets of continuity conditions (with either weak continuity in the state-action pair or strong continuity in the action for each state) on transition kernels. [ABSTRACT FROM AUTHOR]

We consider a semidiscrete finite volume scheme for a degenerate fractional conservation law driven by a cylindrical Wiener process. Making use of the bounded variation (BV) estimates, and a clever adaptation of classical Kružkov theory, we provide estimates on the rate of convergence for approximate solutions to degenerate fractional problems. The main difficulty stems from the degenerate fractional operator and requires a significant departure from the existing strategy to establish Kato's type of inequality. Indeed, recasting the mathematical framework recently developed in Bhauryal et al. (2021, J. Differential Equations , 284 , 433–521), we establish such Kato's type of inequality for a finite volume scheme. Finally, as an application of this theory, we demonstrate numerical convergence rates. [ABSTRACT FROM AUTHOR]

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]

In this work, we study the convergence of a sequence of solutions of degenerate elliptic inclusion problems with variable coercivity and growth exponents. The functional setting involves Lebesgue and Sobolev spaces with variable exponent which varies also with n. [ABSTRACT FROM AUTHOR]

In this paper, we provide some results pertaining to asymptotic behaviour as ε → 0 of the finite-energy sequences of the one-dimensional Cahn-Hilliard functional ... where u ∊ H1(0, 1), and where W is a multi-well potential endowed with a non-standard integrability condition. We introduce a new class of finite-energy sequences, recover its underlying geometric properties as ε → 0, and obtain the related a priori estimates. [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]

We consider purely singular homogeneous Young measures associated with elements of sequences of piecewise constant functions and with limits of such sequences. We first consider a case when the limit of a such sequence is piecewise constant. The next point involves the sequences of bounded oscillating functions, divergent in the strong topology in L∞, but weakly∗ convergent to a homogeneous Young measure. We also present an example of a fast oscillating sequence, illustrating the result. In the presented results, generalizing to some extent known examples, we try to avoid advanced methods of functional analysis that are usually used when solving problems of this type. [ABSTRACT FROM AUTHOR]

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models. [ABSTRACT FROM AUTHOR]

Using the theory of Young measures, we prove the existence of solutions to a strongly quasilinear parabolic system ∂u/ ∂t + A(u) = f where A(u) = -div σ(x, t, u, Du) + σ0(x, t, u, Du0, σ(x, t, u, Du) and σ0 (x, t, u, Du) are satisfy some conditions and f ∈ Lp' (O, T; W-1p' (Ω Rm)). [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]

In this paper we prove the existence of weak solutions for a class of quasilinear parabolic systems, which correspond to diffusion problems, in the form where Ω is a bounded open domain of be given and The function v belongs to is in a moving and dissolving substance, the dissolution is described by f and the motion by g. We prove the existence result by using Galerkin’s approximation and the theory of Young measures.

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 the diffusive limit approximation for a nonlinear radiative heat transfer system that arises in the modeling of glass cooling and greenhouse effects and in astrophysics. The model is considered with the reflective radiative boundary conditions for the radiative intensity and with periodic, Dirichlet and Robin boundary conditions for the temperature. The global existence of weak solutions for this system is given by using a Galerkin method with a careful treatment of the boundary conditions. Using the compactness method, averaging lemma and Young measure theory, we prove our main result that the weak solution converges to a nonlinear diffusion model in the diffusive limit. Moreover, under more regularity conditions on the limit system, the diffusive limit is also analyzed by using a relative entropy method. In particular, we get a rate of convergence. The initial and boundary conditions are assumed to be well-prepared in the sense that no initial or boundary layer exists. [ABSTRACT FROM AUTHOR]

This paper concerns the mathematical justification of a macroscopic Baer–Nunziato PDE bifluid system with a physical relaxation term that is linked to the two viscosities and the two pressure laws of the two compressible phases of the fluid which may be different. This is achieved using an homogenization approach in a periodic framework from a mesoscopic PDE description of two immiscible compressible viscous fluids with interfaces and no mass transfer. Our result extends the work in Bresch D, Hillairet M. [Note on the derivation of multi-component flow systems. Proc Am Math Soc. 2015;143:3429–3443] by allowing to consider different pressure laws for each component introducing an order parameter. This paper is complementary to the recent work [Bresch D, Burtea C, Lagoutière F. Mathematical justification of a compressible bi-fluid system with different pressure laws: a semi-discrete approach and numerical illustrations. Submitted 2021] which focuses on a semi-discretized approach and numerical illustrations. These two papers correspond to the extended versions of the document arXiv:2012.06497. [ABSTRACT FROM AUTHOR]

In this paper we compare the notion of stochastic two-scale convergence in the mean (by Bourgeat, Mikelić and Wright), the notion of stochastic unfolding (recently introduced by the authors), and the quenched notion of stochastic two-scale convergence (by Zhikov and Pyatnitskii). In particular, we introduce stochastic two-scale Young measures as a tool to compare mean and quenched limits. Moreover, we discuss two examples, which can be naturally analyzed via stochastic unfolding, but which cannot be treated via quenched stochastic two-scale convergence. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence of weak solutions for a class of quasilinear parabolic systems, which correspond to diffusion problems, in the ..., where Ω is a bounded open domain of ℝn, 0 < T < ∞ be given and u0 ∈ L²(Ω; ℝm). The function v belongs to Lp' (0, T: W-1,p'(Ω; ℝm)) is in a moving and dissolving substance, the dissolution is described by f and the motion by g. We prove the existence result by using Galerkin's approximation and the theory of Young measures. [ABSTRACT FROM AUTHOR]

We continue considerations concerning Young measures associated with bounded measurable functions from a recent article. We look at them as at the weak measurable, measure-valued mappings. We show examples explaining that we cannot regard a Young measure (i.e. a weak*-measurable mapping) δu(x) as an explicit form of a Young measure associated with a function u. We also consider convergence of the sequences of Young measures. [ABSTRACT FROM AUTHOR]

We study the existence of solutions for the A-Laplace system of the form {-- Δu = f in Ω, u = 0 on ∂ Ω, where -- ΔAu: = -- div(a(|Du|)Du). The existence result is proved by using the technique of Young measures. [ABSTRACT FROM AUTHOR]

The paper deals with the existence of weak solutions to steady quasiNewtonian flows by means of the Galerkin approximations and the measurevalued solutions, namely Young measures, which turned out to be a good tool to describe the weak solutions of our problem in Orlicz spaces. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of weak solutions for obstacle problems. By means of the Young measure theory and a theorem of Kinderlehrer and Stampacchia, we obtain the needed result. [ABSTRACT FROM AUTHOR]

The goal of this paper is to describe the oscillatory microstructure that can emerge from minimizing sequences for nonconvex energies. We consider integral functionals that are defined on real valued (scalar) functions u(x) which are nonconvex in the gradient ∇ u and possibly also in u. To characterize the microstructures for these nonconvex energies, we minimize the associated relaxed energy using two novel approaches: (i) a semianalytical method based on control systems theory, (ii) and a numerical scheme that combines convex splitting together with a modified version of the split Bregman algorithm. These solutions are then used to gain information about minimizing sequences of the original problem and the spatial distribution of microstructure. [ABSTRACT FROM AUTHOR]

We take under consideration Young measures - objects that can be interpreted as generalized solutions of a class of certain nonconvex optimization problems arising among others in nonlinear elasticity or micromagnetics. They can be looked at from several points of view. We look at Young measures as at a class of weak* measurable, measure-valued mappings and consider the basic existence theorem for them. On the basis of this theorem, an imbedding of the set of bounded Borel functions into the set of Young measures is defined. Using the weak* denseness of the set of Young measures associated with simple functions in the set of Young measures, it is shown that this imbedding assigns the Young measure associated with any bounded Borel function. [ABSTRACT FROM AUTHOR]

Nonconvex optimal control problems governed by evolution problems in infinite-dimensional spaces (as, e.g., parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns. [ABSTRACT FROM AUTHOR]

In this article, our aim is to obtain weak solutions for a class of Kirchhoff-type problems − M (∫ Ω A (x , ∇ u) d x) div a (x , ∇ u) = f (x , u) in Ω , by using the theory of Young measures. [ABSTRACT FROM AUTHOR]

We prove the existence of weak solutions to a generalized p-Laplacian systems in degenerate form. The techniques of Young measure for elliptic systems are used to prove the existence result.

In this article we study the existence of weak solutions for quasilinear parabolic system in divergence form with variable growth. By means of Young measures, Galerkin's approximation method and the theory of variable exponents spaces, we obtain the existence of weak solutions.

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]

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that the Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws. [ABSTRACT FROM AUTHOR]

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 $${\rm E}_\infty (u,{\rm {\cal O}})\, : = \, \mathop {{\rm ess}\,\sup}\limits_{\rm {\cal O}} {\rm H}(\cdot, u,{\rm D}u),\quad u\in W_{{\rm loc}}^{1,\infty} (\Omega, {\open R}^N),\quad {\rm {\cal O}}{\Subset}\Omega.$$ The associated PDE system which plays the role of Euler–Lagrange equations in $L^\infty $ is 2 $$\left\{\matrix{{\rm H}_{P}(\cdot, u, {\rm D}u)\, {\rm D}\left({\rm H}(\cdot, u, {\rm D} u)\right) = \, 0, \hfill \cr {\rm H}(\cdot, u, {\rm D} u) \, [\![{\rm H}_{P}(\cdot, u, {\rm D} u)]\!]^\bot \left({\rm Div}\left({\rm H}_{P}(\cdot, u, {\rm D} u)\right)- {\rm H}_{\eta}(\cdot, u, {\rm D} u)\right) = 0,\hfill}\right.$$ where $[\![A]\!]^\bot := {\rm Proj}_{R(A)^\bot }$. Herein we establish that generalised solutions to (2) can be characterised as local minimisers of (1) for appropriate classes of affine variations of the energy. Generalised solutions to (2) are understood as ${\cal D}$ -solutions, a general framework recently introduced by one of the authors. [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]

In the last years measure-valued solutions started to be considered as a relevant notion of solutions if they satisfy the so-called measure-valued – strong uniqueness principle. This means that they coincide with a strong solution emanating from the same initial data if this strong solution exists. This property has been examined for many systems of mathematical physics, including incompressible and compressible Euler system, compressible Navier-Stokes system et al. and there are also some results concerning general hyperbolic systems. Our goal is to provide a unified framework for general systems, that would cover the most interesting cases of systems, and most importantly, we give examples of equations, for which the aspect of measure-valued – strong uniqueness has not been considered before, like incompressible magnetohydrodynamics and shallow water magnetohydrodynamics. [ABSTRACT FROM AUTHOR]

In this paper we initiate the study of second-order variational problems in L ∞ {L^{\infty}} , seeking to minimise the L ∞ {L^{\infty}} norm of a function of the hessian. We also derive and study the respective PDE arising as the analogue of the Euler–Lagrange equation. Given H ∈ C 1 ⁢ (ℝ s n × n) {\mathrm{H}\in C^{1}(\mathbb{R}^{n\times n}_{s})} , for the functional E ∞ ⁢ (u , 𝒪) = ∥ H ⁢ (D 2 ⁢ u) ∥ L ∞ ⁢ (𝒪) , u ∈ W 2 , ∞ ⁢ (Ω) , 𝒪 ⊆ Ω , \mathrm{E}_{\infty}(u,\mathcal{O})=\|\mathrm{H}(\mathrm{D}^{2}u)\|_{L^{\infty}% (\mathcal{O})},\quad u\in W^{2,\infty}(\Omega),\mathcal{O}\subseteq\Omega,{} the associated equation is the fully nonlinear third-order PDE A ∞ 2 ⁢ u := (H X ⁢ (D 2 ⁢ u)) ⊗ 3 : (D 3 ⁢ u) ⊗ 2 = 0. \mathrm{A}^{2}_{\infty}u:=(\mathrm{H}_{X}(\mathrm{D}^{2}u))^{\otimes 3}:(% \mathrm{D}^{3}u)^{\otimes 2}=0.{} Special cases arise when H {\mathrm{H}} is the Euclidean length of either the full hessian or of the Laplacian, leading to the ∞ {\infty} -polylaplacian and the ∞ {\infty} -bilaplacian respectively. We establish several results for (1) and (2), including existence of minimisers, of absolute minimisers and of "critical point" generalised solutions, proving also variational characterisations and uniqueness. We also construct explicit generalised solutions and perform numerical experiments. [ABSTRACT FROM AUTHOR]

We establish a continuity theorem for non-local functionals indexed by Young measures, that we use to deal with homogenization of stochastic non-diffusive reaction differential equations. Non-local effects induced by homogenization of such stochastic differential equations are studied. [ABSTRACT FROM AUTHOR]