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For Ω ⊂ R N open bounded and with a Lipschitz boundary, and 1 ≤ p + ∞ , we consider the Poincare inequality with trace term C p ( Ω ) ‖ u ‖ L p ( Ω ) ≤ ‖ ∇ u ‖ L p ( Ω ; R N ) + ‖ u ‖ L p ( ∂ Ω ) on the Sobolev space W 1 , p ( Ω ) . We show that among all domains Ω with prescribed volume, the constant is minimal on balls. The proof is based on the analysis of a free discontinuity problem.

The paper is concerned with the minimization of the k-th eigenvalue of the Laplace operator with Robin boundary conditions, among all open sets of R N satisfying a volume constraint. We prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets. The main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation. One of the key difficulties (for k ≥ 3 ) comes from the fact that the eigenvalues are critical points.

The purpose of this paper is to analyze regularity properties of local solutions to free discontinuity problems characterized by the presence of multiple phases. The key feature of the problem is related to the way in which two neighboring phases interact: the contact is penalized at jump points, while no cost is assigned to no-jump interfaces which may occur at the zero level of the corresponding state functions. Our main results state that the phases are open and the jump set (globally considered for all the phases) is essentially closed and Ahlfors regular. The proof relies on a multiphase monotonicity formula and on a sharp collective Sobolev extension result for functions with disjoint supports on a sphere, which may be of independent interest.

We give a practical tool to control the L ∞ -norm of the Steklov eigenfunctions in a Lipschitz domain in terms of the norm of the BV-trace operator. The norm of this operator has the advantage to be characterized by purely geometric quantities. As a consequence, we give a spectral stability result for the Steklov eigenproblem under geometric domain perturbations and several examples where stability occurs. In particular we deal with geometric domains which are not equi-Lipschitz, like vanishing holes, merging sets, approximations of inner peaks.

Let [math] be a bounded piecewise [math] open set with convex corners, and let ¶ MS ( u ) : = ∫ Ω | ∇ u | 2 d x + α ℋ 1 ( J u ) + β ∫ Ω | u − g | 2 d x ¶ be the Mumford–Shah functional on the space [math] , where [math] and [math] . We prove that the function [math] such that ¶ − Δ u + β u = β g in Ω , ∂ u ∕ ∂ ν = 0 on ∂ Ω ¶ is a local minimizer of [math] with respect to the [math] -topology. This is obtained as an application of interior and boundary monotonicity formulas for a weak notion of quasiminimizers of the Mumford–Shah energy. The local minimality result is then extended to more general free discontinuity problems taking into account also boundary conditions.

We introduce a new geometric–analytic functional that we analyse in the context of free discontinuity problems. Its main feature is that the geometric term (the length of the jump set) appears with a negative sign. This is motivated by searching quantitative inequalities for the best constants of Sobolev–Poincare inequalities with trace terms in $${\mathbb {R}}^n$$ which correspond to fundamental eigenvalues associated to semilinear problems for the Laplace operator with Robin boundary conditions. Our method is based on the study of this new, degenerate, functional which involves an obstacle problem in interaction with the jump set. Ultimately, this becomes a mixed free discontinuity/free boundary problem occuring above/at the level of the obstacle, respectively.

Pierre Marie Suquet's pioneering work on plasticity paved the way for the mathematical theory of plasticity as we know it today. In this contribution we propose to review the most recent advances on that front and to illustrate how those impact classical problems of quasi-static elasto-plastic evolutions. Most notably, we exhibit new flow rules and derive conditions that prohibit the onset of plastic slips. From these, we obtain new uniqueness results for such evolutions.

We provide a free discontinuity approach to a class of shape optimization problems involving Robin conditions on the free boundary. More precisely, we identify a large family of domains on which such problems are well posed in a way that the extended problem can be considered a relaxed version of the corresponding one on regular domains, we prove existence of a solution and obtain some qualitative information on the optimal sets.

We prove the existence of an optimal partition for the multiphase shape optimization problem which consists in minimizing the sum of the first Robin Laplacian eigenvalue of k mutually disjoint open sets which have a $${\mathcal {H}}^{d-1}$$ -countably rectifiable boundary and are contained into a given box D in $$\mathbb {R}^d$$ .

The uniqueness of the solution to a quasi-static problem in perfect elasto-plasticity is established in the case of a bi-axial test. It is the first example known to us of uniqueness in the context of multi-dimensional elasto-plasticity.

This survey paper is focused on the Saint-Venant inequality for the Laplace operator with Robin boundary conditions. In a larger context, we make the point on the recent advances concerning isoperimetric inequalities of Faber-Krahn type for elastically supported membranes and describe the main ideas of their proofs in both contexts of rearrangement and free discontinuity techniques.

Isoperimetric inequalities for the principal eigenvalues of the Robin-Laplacian are interpreted as free discontinuity problems (of unusual type). We prove a full range of Faber–Krahn inequalities in a nonlinear setting and for non smooth domains, including the open case of the torsional rigidity. The key point of the analysis relies on regularity issues for free discontinuity problems in spaces of functions of bounded variation. As a byproduct, we obtain the best constants for a class of Poincare inequalities with trace terms in \({\mathbb{R}^N}\).

Recent experimental evidence on rubber has revealed that the internal cracks that arise out of the process, often referred to as cavitation, can actually heal. We demonstrate that crack healing can be incorporated into the variational framework for quasistatic brittle fracture evolution that has been developed in the last twenty years. This will be achieved for two-dimensional linearized elasticity in a topological setting, that is, when the putative cracks are closed sets with a preset maximum number of connected components. Other important features of cavitation in rubber, such as near incompressibility and the evolution of the fracture toughness as a function of the cumulative history of fracture and healing, have yet to be addressed even in the proposed topological setting.

The limit behavior of a periodic assembly of a finite number of elasto-plastic phases is investigated as the period becomes vanishingly small. A limit quasi-static evolution is derived through two-scale convergence techniques. It can be thermodynamically viewed as an elasto-plastic model, albeit with an infinite number of internal variables.

In the present paper a sheet of material is considered. It is loaded by uniaxial tensile stress and contains a random distribution of flaw orientations, with the flaws thought of as flat pre-cracks of comparable length, and with all crack planes being oriented perpendicular to the faces of the sheet. Intuition suggests that the most likely flaw to initiate fracture, which will be termed the “most dangerous defect”, lies orthogonally to the major load axis. The purpose of the present paper is to show that such an assumption is incorrect. Neither the most dangerous defect nor the first increments of crack growth will be oriented perpendicularly to the stress direction (nor will they be co-planar with the orientation of the most critical flaw).

In the framework of the energetic approach to rate-independent evolutions, we show that one-dimensional linear perfect plasticity can be obtained by linearization as a variational limit of a finite plasticity model with hardening proposed by A. Mielke (SIAM J. Math. Anal., 2004).

We revisit the homogenization process for a heterogeneous small strain gradient plasticity model considered in [5]. We derive a precise homogenized behavior, independently of any kind of periodicity assumption and demonstrate that it reduces to a model studied in [8] when periodicity is re-introduced.

We prove a stability result for elliptic equations under general Dirichlet–Robin boundary conditions with respect to the variation of the domain under the Hausdorff complementary topology. As a by-product, under the additional assumption of the convergence of the perimeters, we obtain a stability result for the classical Robin–Laplacian.

We prove that the Ciarlet–Nečas non-interpenetration of matter condition can be extended to the case of deformations of hyperelastic brittle materials belonging to the class of special functions of bounded variation (SBV), and can be taken into account for some variational models in fracture mechanics. In order to formulate such a condition, we define the deformed configuration under an SBV map by means of the approximately differentiable representative, and we prove some connected stability results under weak convergence. We provide an application to the case of brittle Ogden materials.

In the context of single-crystal strain gradient plasticity, we focus on the simple shear of a constrained strip in order to study the effects of the material parameters possibly involved in the modelling. The model consists of a deformation theory suggested and left undeveloped by Bardella [(2007). Some remarks on the strain gradient crystal plasticity modelling, with particular reference to the material length scales involved. Int. J. Plasticity 23, 296–322] in which, for each glide, three dissipative length scales are considered; they enter the model through the definition of an effective slip which brings into the isotropic hardening function the relevant plastic strain gradients, averaged by means of a p-norm. By means of the defect energy (i.e., a function of Nye's dislocation density tensor added to the free energy; see, e.g., Gurtin [2002. A gradient theory of single-crystal viscoplasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 50, 5–32]), the model further involves an energetic material length scale. The application suggests that two dissipative length scales may be enough to qualitatively describe the size effect of metals at the microscale, and they are chosen in such a way that the higher-order state variables of the model be the dislocation densities. Moreover, we show that, depending on the crystallography, the size effect governed by the defect energy may be different from what expected (based on the findings of [Bardella, L., 2006. A deformation theory of strain gradient crystal plasticity that accounts for geometrically necessary dislocations. J. Mech. Phys. Solids 54, 128–160] and [Gurtin et al. 2007. Gradient single-crystal plasticity with free energy dependent on dislocation densities. J. Mech. Phys. Solids 55, 1853–1878]), leading mostly to some strengthening. In order to investigate the model capability, we also exploit a Γ -convergence technique to find closed-form solutions in the “isotropic limit”. Finally, we analytically show that in the “perfect plasticity” case, should the dissipative length scales be set to zero, the presence of the sole energetic length scale may lead, as in standard plasticity, to non-uniqueness of solutions.

In this paper we study the crack initiation in a hyper-elastic body governed by a Griffith-type energy. We prove that, during a load process through a time-dependent boundary datum of the type t → tg(x) and in the absence of strong singularities (e.g., this is the case of homogeneous isotropic materials) the crack initiation is brutal, that is, a big crack appears after a positive time ti > 0. Conversely, in the presence of a point x of strong singularity, a crack will depart from x at the initial time of loading and with zero velocity. We prove these facts for admissible cracks belonging to the large class of closed one-dimensional sets with a finite number of connected components. The main tool we employ to address the problem is a local minimality result for the functional \(\epsilon(\nu, \Gamma)\,:=\int_{\Omega} f(x,\nabla v)\,{\rm d}x+k{\mathcal{H}}^{1} (\Gamma),\) where \(\Omega \subseteq {\mathbb{R}}^{2}\) , k > 0 and f is a suitable Caratheodory function. We prove that if the uncracked configuration u of Ω relative to a boundary displacement ψ has at most uniformly weak singularities, then configurations (uΓ, Γ) with \({\mathcal{H}}^{1} (\Gamma)\) small enough are such that \(\epsilon(u,\emptyset) < \epsilon(u_{\Gamma},\Gamma)\) .

A two-phase elasto-plastic material is investigated. It is shown that, if the interface is modeled as the limit of a vanishing layer of a third material, then the resulting two-phase material will exhibit a smaller interfacial dissipation than that of a pure two-phase model.

We study the lower semicontinuity for functionals of the form K → ∫K φ (x, ν)dℋ1 defined on compact sets in ℝ2 with a finite number of connected components and finite ℋ1 measure and apply the result to the study of quasi-static growth of brittle fractures in linearly elastic inhomogeneous and anisotropic bodies.

The existence and multiplicity of geodesics joining two given events in a Lorentzian manifold is studied. The assumptions cover the case of the exterior of a black hole.

Accepté dans Annali della Scuola Normale Superiore di Pisa, Classe di Scienze.; This paper is devoted to prove the existence of strong solutions for a brittle fracture model of quasi-static crack propagation in the two dimensional antiplane setting. As usual, the time continuous evolution is obtained as the limit of a discrete in time evolution by letting the time step tend to zero. The analysis rests on a density lower bound estimate for quasi-minimizers of Mumford-Shah type functionals, under a homogeneous Dirichlet boundary condition on a part of the boundary. In contrast with the previous results, since boundary cracks may be obtained as limits of interior cracks, such a density lower bound has to be established also on balls centered inside the domain but possibly intersecting the Dirichlet boundary. Thanks to a 2D geometrical argument, the discrete in time crack turns out to satisfy a uniform density lower bound which can pass to the limit, leading to the closedness of the continuous in time crack. We also establish better convergence properties of the discrete in time displacement/crack pair towards its time continuous counterpart.

We discuss the energetic formulation of the Gurtin and Anand model (J. Mech. Phys. Solids, 2005) in strain gradient plasticity, and illustrate the related mathematical analysis concerning the existence of quasi-static evolutions.

The elastoplastic quasi-static evolution of a multiphase material—a material with a pointwise varying yield surface and elasticity tensor, together with interfaces between the phases—is revisited in the context of conservative globally minimizing movements. Existence is shown, and classical evolutions are recovered under natural constraints on the plastic dissipation potential. Special attention is paid to the interfaces where the correct dissipation has to be enforced on the interfaces. Further, the evolution is shown to be a limit of that obtained for a model with linear isotropic hardening as the hardening becomes vanishingly small. The duality between plastic strains and admissible stresses is also revisited for Lipschitz boundaries, and its role in deriving a classical evolution is circumscribed. © 2012 Wiley Periodicals, Inc.

Using the tool of two-scale convergence, we provide a rigorous mathematical setting for the homogenization result obtained by Fleck and Willis (J. Mech. Phys. Solids 52 (2004) 1855- 1888) concerning the effective plastic behaviour of a strain gradient composite material. Moreover, moving from deformation theory to flow theory, we prove a convergence result for the homogenization of quasistatic evolutions in the presence of isotropic linear hardening.

The isoperimetric inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions was recently proved by Daners in the context of Lipschitz sets. This paper introduces a new approach to the isoperimetric inequality, based on the theory of special functions of bounded variation (SBV). We extend the notion of the first eigenvalue λ1 for general domains with finite volume (possibly unbounded and with irregular boundary), and we prove that the balls are the unique minimizers of λ1 among domains with prescribed volume.

We consider a class of discrete convex functionals which satisfy a (generalized) coarea formula. These functionals, based on submodular interactions, arise in discrete optimization and are known as a large class of problems which can be solved in polynomial time. In particular, some of them can be solved very efficiently by maximal flow algorithms and are quite popular in the image processing community. We study the limit in the continuum of these functionals, show that they always converge to some "crystalline" perimeter/total variation, and provide an almost explicit formula for the limiting functional.

For p > 1, we prove that all the functions of W 2,p loc (R 2 ) satisfy the Whitney property; i.e., if u ∈ W 2,p loc (R 2 ) is such that ∇u = 0 (in the sense of capacity) on a connected set K C R 2 , then u is constant on K.

We prove that if $\Om \subseteq \R^2$ is bounded and $\R^2 \setminus \Om$ satisfies suitable structural assumptions (for example it has a countable number of connected components), then $W^{1,2}(\Om)$ is dense in $W^{1,p}(\Om)$ for every $1\le p, 24 pages

We propose a time-space discretization of a general notion of quasistatic growth of brittle fractures in elastic bodies proposed in [13] by G. Dal Maso, G.A. Francfort, and R. Toader, which takes into account body forces and surface loads. We employ adaptive triangulations and prove convergence results for the total, elastic and surface energies. In the case in which the elastic energy is strictly convex, we prove also a convergence result for the deformations., 31 pages, 2 figures

We prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a � -convergence result for energies of the form

This chapter presents an advanced dialogue system based on in-car hands-free voice interaction, conceived for obtaining driving assistance and for accessing tourist information while driving. Part of the related activities aimed at developing this “Virtual Intelligent Codriver” are being conducted under the European VICO project. The architecture of the dialogue system is here presented, with a description of its main modules: Front-end Speech Processing, Recognition Engine, Natural Language Understanding, Dialogue Manager and Car Wide Web. The use of a set of HMM recognizers, running in parallel, is being investigated within this project in order to ensure low complexity, modularity, fast response, and to allow a real-time reconfiguration of the language models and grammars according to the dialogue context. A corpus of spontaneous speech interactions was collected at ITC-irst using the Wizard-of-Oz method in a real driving situation. Multiple recognition units specialized on geographical subdomains and simpler language models were experimented using the resulting corpus. This investigation shows that, in presence of large lists of names (e.g. cities, streets, hotels), the choice of the output with maximum likelihood among the active units, although a simple approach, provides better results than the use of a single comprehensive language model.

We propose a discontinuous finite element approximation for a model of quasi-static growth of brittle fractures in linearly elastic bodies formulated by Francfort and Marigo, and based on the classical Griffith's criterion. We restrict our analysis to the case of anti-planar shear and we consider discontinuous displacements which are piecewise affine with respect to a regular triangulation., 34 pages, 4 figures

We define a notion of quasi-static evolution for the elliptic approximation of the Mumford-Shah functional proposed by Ambrosio and Tortorelli. Then we prove that this regular evolution converges to a quasi-static growth of brittle fractures in linearly elastic bodies., 34 pages

By means of a penalization argument due to del Pino and Felmer, we prove the existence of multi-spike solutions for a class of quasilinear elliptic equations under natural growth conditions. Compared with the semilinear case some difficulties arise, mainly concerning the properties of the limit equation. The study of concentration of the solutions requires a somewhat involved analysis in which a Pucci-Serrin type identity plays an important role.

We present the main existence result for quasistatic crack growth in the model proposed by Dal Maso, Francfort, and Toader, and prove some qualitative properties of the solutions.

The interface control domain decomposition (ICDD) method of Discacciati, Gervasio, and Quarteroni [SIAM J. Control Optim., 51 (2013), pp. 3434-3458, doi:10.1137/120890764; J. Coupled Syst. Multiscale Dyn., 1 (2013), pp. 372-392, doi:10.1166/jcsmd.2013.1026] is proposed here to solve the coupling between Stokes and Darcy equations. According to this approach, the problem is formulated as an optimal control problem whose control variables are the traces of the velocity and the pressure on the internal boundaries of the subdomains that provide an overlapping decomposition of the original computational domain. A theoretical analysis is carried out, and the well-posedness of the problem is proved under certain assumptions on both the geometry and the model parameters. An efficient solution algorithm is proposed, and several numerical tests are implemented. Our results show the accuracy of the ICDD method, its computational efficiency, and robustness with respect to the different parameters involved (grid size, polynomial degrees, permeability of the porous domain, thickness of the overlapping region). The ICDD approach turns out to be more versatile and easier to implement than the celebrated model based on the Beavers, Joseph, and Saffman coupling conditions.