Advanced Search

We characterize generalized Young measures, the so-called DiPerna–Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures at the boundary of the domain in the case of the compactification of ℝ by the sphere. We show that this characterization is closely related to the notion of quasiconvexity at the boundary introduced by Ball and Marsden [J.M. Ball and J. Marsden, 86 (1984) 251–277]. As a consequence we get new results on weak ...

We study convergence properties of ${\left\{v\left(\nabla u_k\right)\right\}}_{k\in \mathbb{N}}$ if $v\in C\left({ℝ}^{m\times n}\right)$, $\left|v\left(s\right)\right|\le C(1+|s{|}^p)$, $1<p<+\infty$, has a finite quasiconvex envelope, $u_k\to u$ weakly in $W^{1,p}(\Omega ;{ℝ}^m)$ and for some $g\in C\left(\Omega \right)$ it holds that ${\int }_{\Omega }g\left(x\right)v\left(\nabla u_k\left(x\right)\right)\mathrm{d}x\to {\int }_{\Omega }g\left(x\right)Qv\left(\nabla u\left(x\right)\right)\mathrm{d}x$ as $k\to \infty$. In particular, we give necessary and sufficient conditions for $L^1$-weak convergence of ${\{det\nabla u_k\}}_{k\in \mathbb{N}}$ to $det\nabla u$ if $m=n=p$.

The purpose of this note is to discuss the relationship among Rosenthal's modulus of uniform integrability, Young measures and DiPerna-Majda measures. In particular, we give an explicit characterization of this modulus and state a criterion of the uniform integrability in terms of these measures. Further, we show applications to Fatou's lemma.

We characterize generalized extreme points of compact convex sets. In particular, we show that if the polyconvex hull is convex in ${ℝ}^{m\times n}$, $min(m,n)\le 2$, then it is constructed from polyconvex extreme points via sequential lamination. Further, we give theorems ensuring equality of the quasiconvex (polyconvex) and the rank-1 convex envelopes of a lower semicontinuous function without explicit convexity assumptions on the quasiconvex (polyconvex) envelope.

We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(Ø;{ℝ}^{m\times n})$ if $p\&gt;1$ and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega$ are required and links with lower semicontinuity results...

Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

DiPerna and Majda generalized Young measures so that it is possible to describe “in the limit” oscillation as well as concentration effects of bounded sequences in $L^p$-spaces. Here the complete description of all such measures is stated, showing that the “energy” put at “infinity” by concentration effects can be described in the limit basically by an arbitrary positive Radon measure. Moreover, it is shown that concentration effects are intimately related to rays (in a suitable locally convex geometry)...

DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps ${\left\{u_k\right\}}_{k\in \mathbb{N}}\subset L^p(\Omega ;{ℝ}^m)$ satisfying a linear differential constraint $𝒜u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on $𝒜$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla {\varphi }_k\stackrel{*}{\rightharpoonup }\det \nabla \varphi$ in measures on the closure of $\Omega \subset {ℝ}^n$ if ${\varphi }_k\rightharpoonup \varphi$ in $W^{1,n}(\Omega ;{ℝ}^n)$. This convergence holds, for example, under...

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla u_k\right\}$, bounded in $L^p(\Omega ;{ℝ}^{m\times n})$ if and $\Omega \subset {ℝ}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of are required and links with lower semicontinuity results...