Rate-independent evolution for material models with nonconvexelastic energies is studied without any spatial regularization ofthe inner variable; due to lack of convexity, the model is developedin the framework of Young measures. An existence result for thequasistatic evolution is obtained in terms of compatible systems ofYoung measures. We also show as this result can be equivalentlyreformulated with probabilistic language and leads to thedescription of the quasistatic evolution in terms of stochasticprocesses on a suitable probability space.

We study a non standard unique continuation property for the biharmonic spectral problem $\Delta^2 w=-\lambda\Delta w$ in a 2D corner with homogeneous Dirichlet boundary conditions and a supplementary third order boundary condition on one side of the corner. We prove that if the corner has an angle $0<\theta_0<2\pi$ , $\theta_0\not=\pi$ and $\theta_0\not=3\pi/2$ , a unique continuation property holds. Approximate controllability of a 2-D linear fluid-structure problem follows from this property, with a control acting on the elastic side of a corner in a domain containing a Stokes fluid. The proof of the main result is based in a power series expansion of the eigenfunctions near the corner, the resolution of a coupled infinite set of finite dimensional linear systems, and a result of Kozlov, Kondratiev and Mazya, concerning the absence of strong zeros for the biharmonic operator [Math. USSR Izvestiya 34 (1990) 337–353]. We also show how the same methodologyused here can be adapted to exclude domains with corners to have a localversion of the Schiffer property for the Laplace operator.

In this paper we study the lower semicontinuous envelope with respect tothe L 1-topology of a class of isotropic functionals with lineargrowth defined on mappings from the n-dimensional ball into  ${\mathbb R}^{N}$   that are constrained to take values into a smoothsubmanifold   ${\cal Y}$   of   ${\mathbb R}^{N}$ .

Let $K:=SO\left(2\right)A_1\cup SO\left(2\right)A_2\dots SO\left(2\right)A_{N}$ where $A_1,A_2,\dots, A_{N}$ are matrices of non-zero determinant. Weestablish a sharp relation between the following two minimisationproblems in two dimensions. Firstly the N-well problem with surface energy. Let $p\in\left[1,2\right]$ , $\Omega\subset \mathbb{R}^2$ be a convex polytopal region. Define $$I^p_{\epsilon}\left(u\right)=\int_{\Omega} d^p\left(Du\left(z\right),K\right)+\epsilon\left|D^2u\left(z\right)\right|^2 {\rm d}L^2 z$$ and let A F denote the subspace of functions in $W^{2,2}\left(\Omega\right)$ that satisfy the affine boundary conditionDu=F on $\partial \Omega$ (in the sense of trace), where $F\not\inK$ . We consider the scaling (with respect to ϵ) of $$m^p_{\epsilon}:=\inf_{u\in A_F} I^p_{\epsilon}\left(u\right).$$ Secondly the finite element approximation to the N-well problemwithout surface energy. We will show there exists a space of functions $\mathcal{D}_F^{h}$ whereeach function $v\in \mathcal{D}_F^{h}$ is piecewise affine on a regular(non-degenerate) h-triangulation and satisfies the affine boundarycondition v=lF on $\partial \Omega$ (where l F is affine with $Dl_F=F$ ) such that for $$\alpha_p\left(h\right):=\inf_{v\in \mathcal{D}_F^{h}}\int_{\Omega}d^p\left(Dv\left(z\right),K\right) {\rm d}L^2 z$$ there exists positive constants $\mathcal{C}_1<1<\mathcal{C}_2$ (depending on $A_1,\dots, A_{N}$ , p) for which the following holds true $$\mathcal{C}_1\alpha_p\left(\sqrt{\epsilon}\right)\leq m^p_{\epsilon}\leq\mathcal{C}_2\alpha_p\left(\sqrt{\epsilon}\right) \text{ for all }\epsilon>0.$$

We investigate the regularity of solutions of first order Hamilton-Jacobi equation with super linear growth in the gradient variable. We show that the solutions are locally Hölder continuous with Hölder exponent depending only on the growth of the Hamiltonian. The proof relies on a reverse Hölder inequality.

A unified framework for analyzing the existence of ground states in wideclasses of elastic complex bodies is presented here. The approach makes useof classical semicontinuity results, Sobolev mappings and Cartesiancurrents. Weak diffeomorphisms are used to represent macroscopicdeformations. Sobolev maps and Cartesian currents describe the innersubstructure of the material elements. Balance equations for irregularminimizers are derived. A contribution to the debate about the role of thebalance of configurational actions follows. After describing a list ofpossible applications of the general results collected here, a concretediscussion of the existence of ground states in thermodynamically stablequasicrystals is presented at the end.

In this paper we study asymptotic behaviour of distributed parameter systems governedby partial differential equations (abbreviated to PDE). We first review some recently developed resultson the stability analysis of PDE systems by Lyapunov's second method. On constructing Lyapunov functionalswe prove next an asymptotic exponential stability result for a class of symmetric hyperbolic PDEsystems. Then we apply the result to establish exponential stability of various chemical engineeringprocesses and, in particular, exponential stability of heat exchangers. Through concrete examples weshow how Lyapunov's second method may be extended to stability analysis of nonlinear hyperbolic PDE.Meanwhile we explain how the method is adapted to the framework of Banach spaces Lp , $1<p\leq \infty$ . 


In this paper we study Lavrentiev-type regularization concepts forlinear-quadratic parabolic control problems with pointwise state constraints. Inthe first part, we apply classical Lavrentiev regularization to a problem withdistributed control, whereas in the second part, a Lavrentiev-typeregularization method based on the adjoint operator is applied to boundarycontrol problems with state constraints in the whole domain. The analysis forboth classes of control problems is investigated and numerical tests areconducted. Moreover the method is compared with other numerical techniques.

We present below a new series of conjectures and openproblems in the fields of (global) Optimization and Matrix analysis, in thesame spirit as our recently published paper [J.-B. Hiriart-Urruty, Potpourri of conjectures and open questions in Nonlinear analysis and Optimization. SIAMReview49 (2007) 255–273]. With each problem come a succinct presentation, a list of specificreferences, and a view on the state of the art of the subject.

We are concerned with the asymptotic analysis of optimal controlproblems for 1-D partial differential equations defined on aperiodic planar graph, as the period of the graph tends to zero. Wefocus on optimal control problems for elliptic equations withdistributed and boundary controls. Using approaches of the theory ofhomogenization we show that the original problem on the periodicgraph tends to a standard linear quadratic optimal control problemfor a two-dimensional homogenized system, and its solution can beused as suboptimal controls for the original problem.