Let ${\cal L}$ (x,u,∇ u) be a Lagrangian periodic of period 1 in x 1,...,x n ,u. We shall study the non self intersectingfunctions u: R n ${\to}$ R minimizing ${\cal L}$ ; non self intersecting means that, if u(x 0 + k) + j = u(x 0) for some x 0∈R n and (k , j) ∈Z n × Z, then u(x) = u(x + k) + j $\;\forall$ x. Moser has shown that each of these functions is at finite distance from a plane u = ρ $\cdot$ x and thus has an average slope ρ; moreover, Senn has proven that it is possible to define the average action of u, which is usually called $\beta(\rho)$ since it only depends on the slope of u.Aubry and Senn have noticed a connection between $\beta(\rho)$ and the theory of crystals in ${\bf R}^{n+1}$ , interpreting $\beta(\rho)$ as the energy per area of a crystal face normal to $(-\rho,1)$ . The polar of β is usually called -α; Senn has shown that α is C 1 and that the dimension of the flat of α which contains c depends only on the “rational space” of $\alpha^\prime$ (c). We prove a similar result for the faces (or the faces of the faces, etc.) of the flats of α: they are C 1 and their dimension depends only on the rational space of their normals.

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.

A tracking problem is consideredin the context of a class $\mathcal{S}$ of multi-input,multi-output, nonlinear systems modelled by controlled functionaldifferential equations. The class contains, as a prototype, allfinite-dimensional, linear, m-input, m-output, minimum-phasesystems with sign-definite “high-frequency gain". The first controlobjective is tracking of reference signals r by the output y ofany system in $\mathcal{S}$ : given $\lambda \geq 0$ , construct afeedback strategy which ensures that, for every r (assumed boundedwith essentially bounded derivative) and every system of class $\mathcal{S}$ , the tracking error $e = y-r$ is such that, in the case $\lambda >0$ , $\limsup_{t\rightarrow\infty}\|e(t)\|<\lambda$ or, inthe case $\lambda=0$ , $\lim_{t\rightarrow\infty}\|e(t)\| = 0$ . Thesecond objective is guaranteed output transient performance: theerror is required to evolve within a prescribed performance funnel $\mathcal{F}_\varphi$ (determined by a function φ). Forsuitably chosen functions α, ν and θ, bothobjectives are achieved via a control structure of the form $u(t)=-\nu (k(t))\theta (e(t))$ with $k(t)=\alpha (\varphi(t)\|e(t)\|)$ , whilst maintaining boundedness of the control andgain functions u and k. In the case $\lambda=0$ , the feedbackstrategy may be discontinuous: to accommodate this feature, aunifying framework of differential inclusions is adopted in theanalysis of the general case $\lambda \geq 0$ .

We prove the existence of a positive solution to the BVP $$(\Phi(t)u'(t))'=f(t,u(t)),\,\,\,\,\,\,\,\,\,\,\,u'(0)=u(1)=0, $$ imposing some conditions on Φ and f. In particular, weassume $\Phi(t)f(t,u)$ to be decreasing in t. Our methodcombines variational and topological arguments and can be appliedto some elliptic problems in annular domains. An $L_\infty$ boundfor the solution is provided by the $L_\infty$ norm of any testfunction with negative energy.

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.

Given the probability measure ν over the given region $\Omega\subset \mathbb{R}^n$ , we consider the optimal location of a setΣ composed by n points in Ω in order to minimize theaverage distance $\Sigma\mapsto \int_\Omega \mathrm{dist}\,(x,\Sigma)\,{\rm d}\nu$ (theclassical optimal facility location problem). The paper compares twostrategies to find optimal configurations: the long-term one whichconsists in placing all n points at once in an optimal position, and the short-term one which consists in placing the points one by one addingat each step at most one point and preserving the configurationbuilt at previous steps. We show that the respective optimizationproblems exhibit qualitatively different asymptotic behavior as $n\to\infty$ , although the optimization costs in both cases have the same asymptotic orders of vanishing.


For a fixed bounded open set $\Omega\subset\mathbb{R}^N$ , a sequence of open sets $\Omega_n\subset\Omega$ and a sequence of sets $\Gamma_n\subset\partial\Omega\cap\partial\Omega_n$ , we study theasymptotic behavior of the solution of a nonlinear ellipticsystem posed on $\Omega_n$ , satisfying Neumann boundary conditionson $\Gamma_n$ and Dirichlet boundary conditions on  $\partial\Omega_n\setminus \Gamma_n$ . We obtain a representationof the limit problem which is stable by homogenization and weprove that this representation depends on $\Omega_n$ and $\Gamma_n$ locally.


We are concerned with the Lipschitz modulus of the optimal set mappingassociated with canonically perturbed convex semi-infinite optimizationproblems. Specifically, the paper provides a lower and an upper bound forthis modulus, both of them given exclusively in terms of the problem's data.Moreover, the upper bound is shown to be the exact modulus when the numberof constraints is finite. In the particular case of linear problems theupper bound (or exact modulus) adopts a notably simplified expression. Ourapproach is based on variational techniques applied to certain difference ofconvex functions related to the model. Some results of [M.J. Cánovas et al., J. Optim. Theory Appl. (2008) Online First](which go back to [M.J. Cánovas, J. Global Optim.41 (2008) 1–13] and [Ioffe, Math. Surveys55 (2000) 501–558; Control Cybern.32 (2003) 543–554])constitute the starting point of the present work.

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 y(h)(t,x) be one solution to \[\partial_t y(t,x) - \sum_{i, j=1}^{n}\partial_{j} (a_{ij}(x)\partial_i y(t,x))= h(t,x), \thinspace 0<t<T, \thinspace x\in \Omega\] with a non-homogeneous term h, and $y\vert_{(0,T)\times\partial\Omega} = 0$ ,where $\Omega \subset\Bbb R^n$ is a bounded domain. We discuss an inverse problemof determining n(n+1)/2 unknown functions a ij by $\{ \partial_{\nu}y(h_{\ell})\vert_{(0,T)\times \Gamma_0}$ , $y(h_{\ell})(\theta,\cdot)\}_{1\le \ell\le \ell_0}$ after selecting input sources $h_1, ...,h_{\ell_0}$ suitably, where $\Gamma_0$ is an arbitrary subboundary, $\partial_{\nu}$ denotes the normal derivative, $0 < \theta < T$ and $\ell_0 \in \Bbb N$ . In the case of $\ell_0 = (n+1)^2n/2$ , we provethe Lipschitz stability in the inverse problem if we choose $(h_1, ...,h_{\ell_0})$ from a set ${\cal H} \subset \{ C_0^{\infty}((0,T)\times \omega)\}^{\ell_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$ . Moreover we can take $\ell_0 = (n+3)n/2$ by making special choices for $h_{\ell}$ , $1 \le \ell \le \ell_0$ . The proof is based on a Carlemanestimate.

Motivated by the study of multidimensional control problems of Dieudonné-Rashevsky type, we raise the question how to understand to notion of quasiconvexity for a continuous function f with a convex body K $\subset \mathbb{R}^{nm}$ instead of the whole space $\mathbb{R}^{nm}$ as the range of definition. In the present paper, we trace the consequences of an infinite extension of f outside K, and thus study quasiconvex functions which are allowed to take the value +∞. As an appropriate envelope, we introduce and investigate the lower semicontinuous quasiconvex envelope $f^{(qc)} (v) = {\rm sup} \{ \,g(v)\, \vert \,g : \mathbb{R}^{nm} \rightarrow \mathbb{R} \cup \{ + \infty \}$ quasiconvex and lower semicontinuous, $g(v) \leq f(v) \,\,\,\,\forall v \in \mathbb{R}^{nm}\,\}.$ Our main result is a representation theorem for $f^{({\it qc})}$ which generalizes Dacorogna's well-known theorem on the representation of the quasiconvex envelope of a finite function. The paper will be completed by the calculation of $f^{({\it qc})}$ in two examples.


We apply Robin penalization to Dirichlet optimal control problemsgoverned by semilinear elliptic equations. Error estimates in terms of the penalization parameter are stated. The results are compared with some previous ones in the literature and are checked by a numerical experiment. A detailed study of the regularity of the solutions of the PDEs is carried out.

In this paper we show that a linear variational inequality over an infinite dimensional real Hilbert spaceadmits solutions for every nonempty bounded closed and convex set, if and only if the linear operator involved in the variational inequality is pseudo-monotone in the sense of Brezis.

We consider the Laplacian in a domain squeezedbetween two parallel curves in the plane,subject to Dirichlet boundary conditions on one of the curvesand Neumann boundary conditions on the other.We derive two-term asymptotics for eigenvaluesin the limit when the distance between the curves tends to zero.The asymptotics are uniform and local in the sense thatthe coefficients depend only on the extremal points wherethe ratio of the curvature radii of the Neumann boundaryto the Dirichlet one is the biggest.We also show that the asymptotics can be obtainedfrom a form of norm-resolvent convergencewhich takes into account the width-dependenceof the domain of definition of the operators involved.

We consider an optimal control problem for a system of the form $\dot{x}$ = f(x,u), with a running cost L. We prove an interiorsphere property for the level sets of the corresponding valuefunction V. From such a property we obtain a semiconcavityresult for V, as well as perimeter estimates for the attainablesets of a symmetric control system.

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.

We show how to capture the gradient concentration of the solutions of Dirichlet-type problems subjected to large sources of order ${1\over \sqrt \varepsilon}$ concentrated on an ε-neighborhood of a hypersurface of the domain. To this end we define the gradient Young-concentration measures generated by sequences of finite energy and establish a very simple characterization of these measures.

This paper is concerned with mathematical modelling in the management of a wastewater treatment system. The problem is formulated as looking for a Nash equilibrium of a multiobjective pointwise control problem of aparabolic equation. Existence of solution is proved and a first order optimality system is obtained. Moreover, a numerical method to solve this system is detailed and numerical results are shown in a realistic situation posed in the estuary of Vigo (Spain).


In this note we provide a new geometric lower bound on theso-called Grad's number of a domain Ω in terms of how far Ωis from being axisymmetric. Such an estimate is important in thestudy of the trend to equilibrium for the Boltzmann equation fordilute gases.