3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{u}_{\epsilon }\right)$ bounded in $L^p(\Omega ;{ℝ}^9),1<p<+\infty .$ Here it is shown that, up to a subsequence, $u_{\epsilon }$ may be decomposed as $w_{\epsilon }+z_{\epsilon },$ where $z_{\epsilon }$ carries all the concentration effects, $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_3{w}_{\epsilon }\right)\right|}^p\right\}$ is equi-integrable, and $w_{\epsilon }$ captures the oscillatory behavior, $z_{\epsilon }\to 0$ in measure. In addition, if $\left\{u_{\epsilon }\right\}$ is a recovering sequence then $z_{\epsilon }=z_{\epsilon }\left(x_{\alpha }\right)$ nearby $\partial \Omega .$

This paper is devoted to the study of a turbulent circulation model. Equations are derived from the “Navier-Stokes turbulent kinetic energy” system. Some simplifications are performed but attention is focused on non linearities linked to turbulent eddy viscosity ${\nu }_t$. The mixing length $\ell$ acts as a parameter which controls the turbulent part in ${\nu }_t$. The main theoretical results that we have obtained concern the uniqueness of the solution for bounded eddy viscosities and small values of $\ell$...

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...

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\cdots SO\left(2\right)A_N$ where $A_1,A_2,\cdots ,A_N$ are matrices of non-zero determinant. We establish a sharp relation between the following two minimisation problems in two dimensions. Firstly the -well problem with surface energy. Let $p\in \left[1,2\right]$, $\Omega \subset {ℝ}^2$ be a convex polytopal region. Define $$I_{ϵ}^p\left(u\right)={\int }_{\Omega }{d}^p\left(Du\left(z\right),K\right)+ϵ{\left|D^{2}u\left(z\right)\right|}^2\mathrm{d}{L}^{2}z$$ and let denote the subspace of functions in $W^{2,2}\left(\Omega \right)$ that satisfy the affine boundary condition on $\partial \Omega$ (in the sense of trace), where $F\notin K$. We consider the scaling (with respect to ) of $$m_{ϵ}^p:=\underset{u\in A_F}{inf}{I}_{ϵ}^p\left(u\right).$$ Secondly the finite element approximation to the...