We consider the weak closure WZ of the set Z of all feasible pairs (solution, flow) of the family of potential elliptic systems $$\begin{array}{c}\text{div}\left(\sum _{s=1}^{s_0}{\sigma }_s\left(x\right)F_s^{\text{'}}(\nabla u\left(x\right)+g\left(x\right))-f\left(x\right)\right)=0\text{in}\Omega ,u=(u_1,\cdots ,u_m)\in H_0^1(\Omega ;{𝐑}^m),\sigma =({\sigma }_1,\cdots ,{\sigma }_{s_0})\in S,\end{array}$$ where Ω ⊂ Rn is a bounded Lipschitz domain, Fs are strictly convex smooth functions with quadratic growth and $S=\{\sigma measurable\mid {\sigma }_s\left(x\right)=0\text{or}1,s=1,\cdots ,s_0,{\sigma }_1\left(x\right)+\cdots +{\sigma }_{s_0}\left(x\right)=1\}$. We show that WZ is the zero level set for an integral functional with the integrand $Qℱ$ being the A-quasiconvex envelope for a certain function $ℱ$ and the operator A = (curl,div)m. If the functions Fs are isotropic, then on the characteristic cone...

In this paper, we consider a Borel measurable function on the space of $m\times n$ matrices $f:M^{m\times n}\to \overline{ℝ}$ taking the value $+\infty$, such that its rank-one-convex envelope $Rf$ is finite and satisfies for some fixed $p>1$: $$-c_0\le Rf\left(F\right)\le {c(1+\parallel F\parallel }^p)\text{for}\text{all}F\in M^{m\times n},$$ where $c,c_0>0$. Let $Ø$ be a given regular bounded open domain of ${ℝ}^n$. We define on $W^{1,p}(Ø;{ℝ}^m)$ the functional $$I\left(u\right)={\int }_{Ø}f\left(\nabla u\left(x\right)\right)dx.$$ Then, under some technical restrictions on $f$, we show that the relaxed functional $\overline{I}$ for the weak topology of $W^{1,p}(Ø;{ℝ}^m)$ has the integral representation: $$\overline{I}\left(u\right)={\int }_{Ø}Q\left[Rf\right]\left(\nabla u\left(x\right)\right)dx,$$ where for a given function $g$, $Qg$ denotes its quasiconvex...

Multidimensional vectorial non-quasiconvex variational problems are relaxed by means of a generalized-Young-functional technique. Selective first-order optimality conditions, having the form of an Euler-Weiestrass condition involving minors, are formulated in a special, rather a model case when the potential has a polyconvex quasiconvexification.

A class of parabolic initial-boundary value problems is considered, where admissible coefficients are given in certain intervals. We are looking for maximal values of the solution with respect to the set of admissible coefficients. We give the abstract general scheme, proposing how to solve such problems with uncertain data. We formulate a general maximization problem and prove its solvability, provided all fundamental assumptions are fulfilled. We apply the theory to certain Fourier obstacle type...

The existence of a solution to the dynamic contact of a body having a singular memory with a rigid undeformable support is proved under some weaker assumption than that in [3].

We present some consequences of a deep result of J. Lindenstrauss and D. Preiss on $\Gamma$-almost everywhere Fréchet differentiability of Lipschitz functions on $c_0$ (and similar Banach spaces). For example, in these spaces, every continuous real function is Fréchet differentiable at $\Gamma$-almost every $x$ at which it is Gâteaux differentiable. Another interesting consequences say that both cone-monotone functions and continuous quasiconvex functions on these spaces are $\Gamma$-almost everywhere Fréchet differentiable....

We compute the quasiconvex envelope of certain functions defined on the space $M_{mn}$ of real $m\times n$ matrices. These functions are basically functions of a quadratic form on $M_{mn}$. The quasiconvex envelope computation is applied to densities that are related to the James-Ericksen elastic stored energy function.

In compressible Neohookean elasticity one minimizes functionals which are composed by the sum of the $L^2$ norm of the deformation gradient and a nonlinear function of the determinant of the gradient. Non–interpenetrability of matter is then represented by additional invertibility conditions. An existence theory which includes a precise notion of invertibility and allows for cavitation was formulated by Müller and Spector in 1995. It applies, however, only if some $L^p$-norm of the gradient with $p\>2$ is controlled...

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit superior of the L∞ norms of gradient fields. The...

We examine the composition of the L∞ norm with weakly convergent sequences of gradient fields associated with the homogenization of second order divergence form partial differential equations with measurable coefficients. Here the sequences of coefficients are chosen to model heterogeneous media and are piecewise constant and highly oscillatory. We identify local representation formulas that in the fine phase limit provide upper bounds on the limit...