The theory of compensated compactness of Murat and Tartar links the algebraic condition of rank-r convexity with the analytic condition of weak lower semicontinuity. The former is an algebraic condition and therefore it is, in principle, very easy to use. However, in applications of this theory, the need for an efficient classification of rank-r convex forms arises. In the present paper, we define the concept of extremal 2-forms  and characterize them in the rotationally invariant jointly rank-r convex case.

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.

We study a variational problem which was introduced by Hannon,Marcus and Mizel [ESAIM: COCV9 (2003) 145–149] todescribe step-terraces on surfaces of so-called “unorthodox” crystals.We show that there is no nondegenerate intervals on which the absolutevalue of a minimizer is $\pi/2$ identically.

In this paper, weconsider probability measures μ and ν on a d-dimensionalsphere in ${\bf R}^{d+1}, d \geq 1,$ and cost functions of the form $c({\bf x},{\bf y})=l(\frac{|{\bf x}-{\bf y}|^2}{2})$ that generalize those arising in geometric optics where $l(t)=-\log t.$ We prove that if μ and ν vanish on $(d-1)$ -rectifiable sets,if |l'(t)|>0, $\lim_{t\rightarrow 0^+}l(t)=+\infty,$ and $g(t):=t(2-t)(l'(t))^2$ is monotone then thereexists a unique optimal map T o that transports μ onto $\nu,$ whereoptimality is measured against c. Furthermore, $\inf_{{\bf x}}|T_o{\bf x}-{\bf x}|>0.$ Our approach is based on direct variational arguments.In the special case when $l(t)=-\log t,$ existence of optimal maps on thesphere was obtained earlier in [Glimm and Oliker, J. Math. Sci.117 (2003) 4096-4108]and [Wang, Calculus of Variations and PDE's20 (2004) 329-341] under more restrictive assumptions. In these studies, it was assumed that either μ and ν are absolutelycontinuous with respect to the d-dimensional Haussdorff measure, or they have disjoint supports. Another aspect of interestin this work is that it is in contrast with the work in[Gangbo and McCann, Quart. Appl. Math.58 (2000) 705-737] where it is proved that when l(t)=t thenexistence of an optimal map fails when μ and ν are supported by Jordan surfaces.

We study an homogenization problem for Hamilton-Jacobi equations in the geometry of Carnot Groups. The tiling and the corresponding notion of periodicity are compatible with the dilatations of the Group and use the Lie bracket generating property.

In this paper we prove that every weakand strong localminimizer $u\in{W^{1,2}(\Omega,\mathbb{R}^3)}$ of the functional $I(u)=\int_\Omega|Du|^2+f({\rm Adj}Du)+g({\rm det}Du),$ where $ u:\Omega\subset\mathbb{R}^3\to \mathbb{R}^3$ , f grows like $|{\rm Adj}Du|^p$ , g growslike $|{\rm det}Du|^q$ and1<q<p<2, is $C^{1,\alpha}$ on an opensubset $\Omega_0$ of Ω such that ${\it meas}(\Omega\setminus \Omega_0)=0$ . Suchfunctionals naturally arise from nonlinear elasticity problems. The keypoint in order to obtain the partial regularity result is toestablish an energy estimate of Caccioppoli type, which is based onan appropriate choice of the test functions. The limit case $p=q\le 2$ is also treated for weak local minimizers.

We approximate, in the sense ofΓ-convergence, free-discontinuity functionals with lineargrowth in the gradient by a sequence of non-local integralfunctionals depending on the average of the gradients on smallballs. The result extends to higher dimension what we already proved inthe one-dimensional case.

This work deals with a two-dimensional inverse problem in the field of tomography. The geometry of an unknown inclusion has to be reconstructed from boundary measurements. In this paper, we extend previous results of R. Kress and his coauthors: the leading idea is to use the conformal mapping function as unknown. We establish an integrodifferential equation that the trace of the Riemann map solves. We write it as a fixed point equation and give conditions for contraction. We conclude with a series of numerical examples illustrating the performance of the method.

We are concerned with the optimal control of a nonlinear stochastic heat equation on a bounded real interval with Neumann boundary conditions. The specificity here is that both the control and the noise act on the boundary. We start by reformulating the state equation as an infinite dimensional stochastic evolution equation. The first main result of the paper is the proof of existence and uniqueness of a mild solution for the corresponding Hamilton-Jacobi-Bellman (HJB) equation. The C 1 regularity of such a solution is then used to construct the optimal feedback for the control problem. In order to overcome the difficulties arising from the degeneracy of the second order operator and from the presence of unbounded terms we study the HJB equation by introducing a suitable forward-backward system of stochastic differential equations as in the appraoch proposed in [Fuhrman and Tessitore, Ann. Probab.30 (2002) 1397-1465; Pardoux and Peng, Lect. Notes Control Inf. Sci.176 (1992) 200-217] for finite dimensional and infinite dimensional semilinear parabolic equations respectively.