We study the variational problem for $N$-parallel curves on a Finsler surface by means of exterior differential systems using Griffiths’ method. We obtain the conditions when these curves are extremals of a length functional and write the explicit form of Euler–Lagrange equations for this type of variational problem.

In this paper we propose and analyze stable variational formulations for convection diffusion problems starting from concepts introduced by Sangalli. We derive efficient and reliable a posteriori error estimators that are based on these formulations. The analysis of resulting adaptive solution concepts, when specialized to the setting suggested by Sangalli’s work, reveals partly unexpected phenomena related to the specific nature of the norms induced by the variational formulation. Several remedies, based on other specifications, are explored and illustrated by numerical experiments.

The Stefan problem is coupled with a spatially inhomogeneous and anisotropic Gibbs–Thomson condition at the phase boundary. We show the long-time existence of weak solutions for the non-degenerate Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law and a conditional existence result for the corresponding degenerate Stefan problem. To this end, approximate solutions are constructed by means of variational problems for energy functionals with spatially inhomogeneous and anisotropic interfacial energy. By passing to the limit, we establish solutions of the Stefan problem with a spatially inhomogeneous and anisotropic Gibbs–Thomson law in a weak generalised BV-formulation.

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.

The present article is an overview of some mathematical results, whichprovide elements of rigorous basis for some multiscalecomputations in materials science. The emphasis is laid upon atomisticto continuum limits for crystalline materials. Various mathematicalapproaches are addressed. Thesetting is stationary. The relation to existing techniques used in the engineeringliterature is investigated.

In order to describe a solid which deforms smoothly in some region, but non smoothly in some other region, many multiscale methods have recently been proposed. They aim at coupling an atomistic model (discrete mechanics) with a macroscopic model (continuum mechanics). We provide here a theoretical ground for such a coupling in a one-dimensional setting. We briefly study the general case of a convex energy, and next concentrate on a specific example of a nonconvex energy, the Lennard-Jones case. In the latter situation, we prove that the discretization needs to account in an adequate way for the coexistence of a discrete model and a continuous one. Otherwise, spurious discretization effects may appear. We provide a numerical analysis of the approach.

We consider symmetries of the Dedonder equation arising from variational problems with partial derivatives. Assuming a proper action of the symmetry group, we identify a set of reduced equations on an open dense subset of the domain of definition of the fields under consideration. By continuity, the Dedonder equation is satisfied whenever the reduced equations are satisfied.

This paper presents the numerical analysis for a variational formulation of rate-independent phase transformations in elastic solids due to Mielke et al. The new model itself suggests an implicit time-discretization which is combined with thefinite element method in space. A priori error estimates are established for the quasioptimal spatial approximation of the stress fieldwithin one time-step. A posteriorierror estimates motivate an adaptive mesh-refining algorithm for efficient discretization. The proposed scheme enables numerical simulations which show that the model allows for hysteresis.

This article is devoted to the study of a perturbation with a viscosity termin an elliptic equation involving the p-Laplacian operator and related tothe best contant problem in Sobolev inequalities in the critical case.We prove first that this problem, together with the equation, is stableunder this perturbation, assuming some conditions on the datas. In thenext section, we show that the zero solution is strongly isolated in somesense, among the space of the solutions. Actually, we end the paper bygiving some analoguous results in the case where the datas presentsymmetries.