We prove that the critical points of the 3d nonlinear elasticity functionalon shells of small thickness h and around the mid-surface S of arbitrary geometry, converge as h → 0to the critical points of the vonKármán functional on S, recently proposed in [Lewicka et al., Ann. Scuola Norm. Sup. Pisa Cl. Sci. (to appear)].This result extends the statement in [Müller and Pakzad, Comm. Part. Differ. Equ. 33 (2008) 1018–1032], derived for the case of plates when $S\subset\mathbb{R}^2$.The convergence holds provided the elastic energies of the 3d deformations scale like h4 and the external body forces scale like h3.

This paper gives a rigorous derivationof a functional proposed by Aftalion and Rivière [Phys. Rev. A64 (2001) 043611] to characterize the energy of vortex filamentsin a rotationally forced Bose-Einstein condensate. Thisfunctional is derived as a Γ-limitof scaled versions of the Gross-Pitaevsky functional for the wave function of such a condensate. In most situations, the vortex filament energy functional is either unbounded below or has only trivial minimizers, but we establish the existence of large numbers of nontrivial local minimizers and we prove that, given any suchlocal minimizer, the Gross-Pitaevsky functionalhas a local minimizer that is nearby (in a suitable sense) whenever a scalingparameter is sufficiently small.

Necessary conditions of optimality in the form ofUnmaximized Inclusions (UI) are derived for optimal controlproblems with state constraints. The conditions presented heregeneralize earlier optimality conditions to problems that may benonconvex. The derivation of UI-type conditions in the absence of the convexity assumption is of particularimportance when deriving necessary conditions for constrainedproblems. We illustrate this feature by establishing, as anapplication, optimality conditions for problems that in additionto state constraints incorporate mixed state-control constraints.

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacityof convex bodies, we discuss the role of concavity inequalities in shape optimization, andwe provide several counterexamples to the Blaschke-concavity of variational functionals,including capacity. We then introduce a new algebraic structure on convex bodies, whichallows to obtain global concavity and indecomposability results, and we discuss theirapplication to isoperimetric-like inequalities. As a byproduct of this approach we alsoobtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class offunctionals involving Dirichlet energies and the surface measure, we perform a localanalysis of strictly convex portions of the boundary via second ordershape derivatives. This allows in particular to exclude the presence of smooth regionswith positive Gauss curvature in an optimal shape for Pólya-Szegö problem.

We prove Lipschitz continuity for local minimizers of integral functionals of the Calculus of Variations in the vectorial case, where the energy density depends explicitly on the space variables and has general growth with respect to the gradient. One of the models is $$ F\left(u \right)=\int_{\Omega}a(x)[h\left(|Du|\right)]^{p(x)}{\rm d}x $$ with h a convex function with general growth (also exponential behaviour is allowed).

In this article, the homogenization process of periodic structures is analyzed using Bloch waves in the case of system of linear elasticity in three dimensions. The Bloch wave method for homogenization relies on the regularity of thelower Bloch spectrum. For the three dimensional linear elasticity system,the first eigenvalue is degenerate of multiplicity three and henceexistence of such a regular Bloch spectrum is not guaranteed. Theaim here is to develop all necessary spectral tools to overcome thesedifficulties. The existence of a directionally regular Bloch spectrum isproved and isused in the homogenization. As a consequence an interesting relation betweenhomogenization process and wave propagation in the homogenized medium isobtained. Existence of a spectral gap for the directionally regular Bloch spectrum is established and as a consequenceit is proved that higher modes apart from the first three do not contribute to the homogenization process.

In this note we prove compactness for the Cahn–Hilliard functional without assumingcoercivity of the multi-well potential.

The liner parabolic equation \hbox{$\frac{\pp y}{\pp t}-\frac12\,\D y+F\cdot\na y={\vec{1}}_{\calo_0}u$}$\frac{\mathit{\partial y}}{\mathit{\partial t}}\mathrm{-}\frac{\mathrm{1}}{\mathrm{2}}\hspace{0.17em}\mathit{\Delta y}\mathrm{+}\mathit{F}\mathrm{·}\mathrm{\nabla }\mathit{y}\mathrm{=}{{1}}_{{\mathrm{𝒪}}_{\mathrm{0}}}\mathit{u}$ with Neumann boundary condition on a convex open domain 𝒪 ⊂ ℝdwith smooth boundary is exactly null controllable on each finite interval if 𝒪0is an open subset of 𝒪which contains a suitable neighbourhood of the recession cone of \hbox{$\ov\calo$}$\overline{\mathrm{𝒪}}$. Here, F : ℝd → ℝd is a bounded, C1-continuous function, and F = ∇g, where g is convex and coercive.

We consider an evolution equation similar to that introduced by Vese in [Comm.Partial Diff. Eq. 24 (1999) 1573–1591] and whose solutionconverges in large time to the convex envelope of the initial datum. We give a stochasticcontrol representation for the solution from which we deduce, under quite generalassumptions that the convergence in the Lipschitz norm is in fact exponential in time.

This paper is part of a larger project initiated with [2]. The final aim of the present paper is to give bounds for the homogenized (or effective) conductivity in two dimensional linear conductivity. The main focus is therefore the periodic setting. We prove new variational principles that are shown to be of interest in finding bounds on the homogenized conductivity. Our results unify previous approaches by the second author and make transparent the central role of quasiconformal mappings in all the two dimensional G-closure problems in conductivity.

This paper is concerned with the construction of local observers for nonlinear systems without inputs, satisfying an observability rank condition. The aim of this study is, first, to define an homogeneous approximation that keeps the observability property unchanged at the origin. This approximation is further used in the synthesis of a local observer which is proven to be locally convergent for Lyapunov-stable systems. We compare the performance of the homogeneous approximation observer with the classical linear approximation observer on an example.

This paper addresses a new differential game problem with forward-backward doubly stochastic differential equations. There are two distinguishing features. One is that our game systems are initial coupled, rather than terminal coupled. The other is that the admissible control is required to be adapted to a subset of the information generated by the underlying Brownian motions. We establish a necessary condition and a sufficient condition for an equilibrium point of nonzero-sum games and a saddle point of zero-sum games. To illustrate some possible applications, an example of linear-quadratic nonzero-sum differential games is worked out. Applying stochastic filtering techniques, we obtain an explicit expression of the equilibrium point.

Homogenization of integral functionals is studiedunder the constraint that admissible maps have to take their valuesinto a given smooth manifold. The notion of tangentialhomogenization is defined by analogy with the tangentialquasiconvexity introduced by Dacorogna et al. [Calc. Var. Part. Diff. Eq. 9 (1999) 185–206]. For energies with superlinear or linear growth, aΓ-convergence result is established in Sobolev spaces, thehomogenization problem in the space of functions of boundedvariation being the object of [Babadjian and Millot, Calc. Var. Part. Diff. Eq.36 (2009) 7–47].

The Euler−Poinsot rigid bodymotion is a standard mechanical system and it is a model for left-invariant Riemannianmetrics on SO(3). In this article using theSerret−Andoyer variables weparameterize the solutions and compute the Jacobi fields in relation with the conjugatelocus evaluation. Moreover, the metric can be restricted to a 2D-surface, and theconjugate points of this metric are evaluated using recent works on surfaces ofrevolution. Another related 2D-metric on S2 associated to the dynamics of spin particles withIsing coupling is analysed using both geometric techniques and numerical simulations.

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 study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of this problem in numerical examples.

We show that the set of nonnegative equilibrium-like states, namely, like $ (y_d, 0) $ of the semilinear vibrating string that can be reached from any non-zero initial state $ (y_0, y_1) \in H^1_0 (0,1) \times L^2 (0,1)$ , by varying its axial load and the gain of damping, is dense in the “nonnegative” part of the subspace $ L^2 (0,1) \times \{0\} $ of $ L^2 (0,1) \times H^{-1} (0,1)$ . Our main results deal with nonlinear terms which admit at most the linear growth at infinity in $ \; y \; $ and satisfy certain restriction on their total impact on (0,∞) with respect to the time-variable.

We formulate an Hamilton-Jacobi partial differential equation H( x, D u(x))=0 on a n dimensional manifold M, with assumptions of convexity of H(x, .) and regularity of H (locally in a neighborhood of {H=0} in T*M); we define the “minsol solution” u, a generalized solution; to this end, we view T*Mas a symplectic manifold.The definition of “minsol solution” is suited to proving regularity results about u; in particular, we provein the first part that the closure of the set where u is not regular may be covered by a countable number of $n-1$ dimensional manifolds, but for a ${{\mathcal H}}^{n-1}$ negligeable subset.These results can be applied to the cutlocus of a C 2 submanifold of a Finsler manifold.

We describe precisely, under generic conditions, the contact of the accessibility set at time T with an abnormal direction, first for a single-input affine control system with constraint on the control, and then as an application for a sub-Riemannian system of rank 2. As a consequence we obtain in sub-Riemannian geometry a new splitting-up of the sphere near an abnormal minimizer γ into two sectors, bordered by the first Pontryagin's cone along γ, called the L∞-sector and the L2-sector. Moreover we find again necessary and sufficient conditions of optimality of an abnormal trajectory for such systems, for any optimization problem.