In this paper we study the lower semicontinuous envelope with respect to the L1-topology of a class of isotropic functionals with linear growth defined on mappings from the n-dimensional ball into  ${ℝ}^N$  that are constrained to take values into a smooth submanifold  $𝒴$  of  ${ℝ}^N$.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an Lp-space (p < ∞). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider control problems governed by semilinear parabolic equations with pointwise state constraints and controls in an $L^p$-space ($p\&lt;\infty$). We construct a correct relaxed problem, prove some relaxation results, and derive necessary optimality conditions.

We consider the weak closure $WZ$ of the set $Z$ of all feasible pairs (solution, flow) of the family of potential elliptic systems$$$$where $\Omega \subset {𝐑}^n$ is a bounded Lipschitz domain, $F_s$ 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 $𝐀={\left(\text{curl,div}\right)}^m$. If the functions $F_s$ are isotropic, then on the characteristic cone $\Lambda$ (defined by the operator $𝐀$) $Qℱ$ coincides...

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.

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 $p\&gt;2$ 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...

Viene studiata la semicontinuità rispetto alla topologia di $L^{\mathrm{\infty }}(\mathrm{\Omega };{𝐑}^m)$ per alcuni funzionali del Calcolo delle Variazioni dipendenti da funzioni a valori vettoriali.

Γ):Γ ∈ &#x1d49c;, ℋ1(Γ) = l}, where ℋ1D1,...,Dk }  ⊂ Rd . The cost functional ℰ(Γ) is the Dirichlet energy of Γ defined through the Sobolev functions on Γ vanishing on the points Di. We analyze the existence of a solution in both the families of connected sets and of metric graphs. At the end, several explicit examples are discussed.