Young measures, superposition and transport

We discuss a space of Young measures in connection with some variational problems. We use it to present a proof of the Theorem of Tonelli on the existence of minimizing curves. We generalize a recent result of Ambrosio, Gigli and Savar\'e on the decomposition of the weak solutions of the transport equation. We also prove, in the context of Mather theory, the equality between Closed measures and Holonomic measures

Young measures supported on invertible matrices

Motivated by variational problems in nonlinear elasticity depending on the deformation gradient and its inverse, we completely and explicitly describe Young measures generated by matrix-valued mappings \{Y_k\}_{k\in\N} \subset L^p(\O;\R^{n\times n}), \O\subset\R^n, such that \{Y_k^{-1}\}_{k\in\N} \subset L^p(\O;\R^{n\times n}) is bounded, too. Moreover, the constraint $\det Y_k>0$ can be easily included and is reflected in a condition on the support of the measure. This condition typically occurs in problems of nonlinear-elasticity theory for hyperelastic materials if $Y:=\nabla y$ for y\in W^{1,p}(\O;\R^n). Then we fully characterize the set of Young measures generated by gradients of a uniformly bounded sequence in W^{1,\infty}(\O;\R^n) where the inverted gradients are also bounded in L^\infty(\O;\R^{n\times n}). This extends the original results due to D. Kinderlehrer and P. Pedregal

Liftings, Young measures, and lower semicontinuity

This work introduces liftings and their associated Young measures as new tools to study the asymptotic behaviour of sequences of pairs $(u_j,Du_j)j$ for $(u_j)_j \in \mathrm{BV}(\Omega;\mathbb{R}^m)$ under weak* convergence. These tools are then used to prove an integral representation theorem for the relaxation of the functional $$\mathcal{F}\colon u\to\int_\Omega f(x,u(x),\nabla u(x)) \;\mathrm{dx},\quad u\in\mathrm{W}^{1,1}({\Omega};\mathbb{R}^m),\quad {\Omega}\in\mathbb{R}^d\text{ open},$$ to the space $\mathrm{BV}(\Omega; \mathbb{R}^m)$. Lower semicontinuity results of this type were first obtained by Fonseca and M\"uller [Arch. Ration. Mech. Anal. 123 (1993), 1-49] and later improved by a number of authors, but our theorem is valid under more natural, essentially optimal, hypotheses than those currently present in the literature, requiring principally that $f$ be Carath\'eodory and quasiconvex in the final variable. The key idea is that liftings provide the right way of localising $\mathcal{F}$ in the $x$ and $u$ variables simultaneously under weak* convergence. As a consequence, we are able to implement an optimal measure-theoretic blow-up procedure.Comment: 75 pages. Updated to correct a series of minor typos/ inaccuracies. The statement and proof of Theorem have also been amended- subsequent steps relying upon the Theorem did not require updatin

Young measures, Cartesian maps, and polyconvexity

We consider the variational problem consisting of minimizing a polyconvex integrand for maps between manifolds. We offer a simple and direct proof of the existence of a minimizing map. The proof is based on Young measures

Young measures in a nonlocal phase transition problem

A nonlocal variational problem modelling phase transitions is studied in the framework of Young measures. The existence of global minimisers among functions with internal layers on an infinite tube is proved by combining a weak convergence result for Young measures and the principle of concentration-compactness. The regularity of such global minimisers is discussed, and the nonlocal variational problem is also considered on asymptotic tubes

Young Measures Generated by Ideal Incompressible Fluid Flows

In their seminal paper "Oscillations and concentrations in weak solutions of the incompressible fluid equations", R. DiPerna and A. Majda introduced the notion of measure-valued solution for the incompressible Euler equations in order to capture complex phenomena present in limits of approximate solutions, such as persistence of oscillation and development of concentrations. Furthermore, they gave several explicit examples exhibiting such phenomena. In this paper we show that any measure-valued solution can be generated by a sequence of exact weak solutions. In particular this gives rise to a very large, arguably too large, set of weak solutions of the incompressible Euler equations.Comment: 35 pages. Final revised version. To appear in Arch. Ration. Mech. Ana