Existence and iterative approximation of solutions of some systems of variational inequalities and inclusions

In 1968, Brézis [Ann. Inst. Fourier (Grenoble), 18 (1) (1968) 115-175] initiated the study of the existence theory of a class of variational inequalities later known as variational inclusions, using proximal-point mappings due to Moreau [Bull. Soc. Math. France, 93 (1965) 273-299]. Variational inclusions include variational, quasi-variational, variational-like inequalities as special cases. In 1985, Pang [Math. Prog. 31 (1985) 206-219] showed that a variety of equilibrium models can be uniformly modelled as a variational inequality defined on the product sets equivalent to a system of variational inequalities and discuss the convergence of method of decomposition for system of variational inequalities. Motivated by the recent research work in this directions, we consider some systems of variational (-like) inequalities and inclusions; develop the iterative algorithms for finding the approximate solutions and discuss their convergence criteria. Further, we study the sensitivity analysis of solution of the system of variational inclusions. The techniques and results presented here improve the corresponding techniques and results for the variational inequalities and inclusions in the literature. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Two row mixed-integer cuts via lifting

Recently Andersen et al. [1], Borozan and Cornuéjols [6] and Cornuéjols and Margot [9] have characterized the extreme valid inequalities of a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. These inequalities are either split cuts or intersection cuts derived using maximal lattice-free convex sets. In order to use these inequalities to obtain cuts from two rows of a general simplex tableau, one approach is to extend the system to include all possible non-negative integer variables (giving the two row mixed-integer infinite-group problem), and to develop lifting functions giving the coefficients of the integer variables in the corresponding inequalities. In this paper, we study the characteristics of these lifting functions. We show that there exists a unique lifting function that yields extreme inequalities when starting from a maximal lattice-free triangle with multiple integer points in the relative interior of one of its sides, or a maximal lattice-free triangle with integral vertices and one integer point in the relative interior of each side. In the other cases (maximal lattice-free triangles with one integer point in the relative interior of each side and non-integral vertices, and maximal lattice-free quadrilaterals), non-unique lifting functions may yield distinct extreme inequalities. For the latter family of triangles, we present sufficient conditions to yield an extreme inequality for the two row mixed-integer infinite-group problem.     

Inégalités de Poincaré cinétiques

In this Note we prove Poincaré type inequalities for a family of kinetic equations. We apply this inequality to the variational solution of a linear kinetic model by generalizing the STILS method (Azerad, 1996 [1]; Azerad and Pousin, 1996 [2]) to a kinetic setting.     

Modified logarithmic Sobolev inequalities and transportation inequalities

We present a class of modified logarithmic Sobolev inequality, interpolating between Poincaré and logarithmic Sobolev inequalities, suitable for measures of the type exp (−|x|^α) or exp (−|x|^α log ^β(2+|x|)) (α ∈]1,2[ and β ∈ ℝ) which lead to new concentration inequalities. These modified inequalities share common properties with usual logarithmic Sobolev inequalities, as tensorisation or perturbation, and imply as well Poincaré inequality. We also study the link between these new modified logarithmic Sobolev inequalities and transportation inequalities. Send offprint requests to: Ivan Gentil     

Self-improvement of Poincaré Type Inequalities Associated with Approximations of the Identity and Semigroups

The purpose of this paper is to present a general method that allows us to study self-improving properties of generalized Poincaré inequalities. When measuring the oscillation in a given cube, we replace the average by an approximation of the identity or a semigroup scaled to that cube and whose kernel decays fast enough. We apply the method to obtain self-improvement in the scale of Lebesgue spaces of Poincaré type inequalities. In particular, we propose some expanded Poincaré estimates that take into account the lack of localization of the approximation of the identity or the semigroup. As a consequence of this method we are able to obtain global pseudo-Poincaré inequalities.     

On the set covering polyhedron of circulant matrices

A well known family of minimally nonideal matrices is the family of the incidence matrices of chordless odd cycles. A natural generalization of these matrices is given by the family of circulant matrices. Ideal and minimally nonideal circulant matrices have been completely identified by Cornuéjols and Novick [G. Cornuéjols, B. Novick, Ideal 0 - 1 matrices, Journal of Combinatorial Theory B 60 (1994) 145–157]. In this work we classify circulant matrices and their blockers in terms of the inequalities involved in their set covering polyhedra. We exploit the results due to Cornuéjols and Novick in the above-cited reference for describing the set covering polyhedron of blockers of circulant matrices. Finally, we point out that the results found on circulant matrices and their blockers present a remarkable analogy with a similar analysis of webs and antiwebs due to Pêcher and Wagler [A. Pêcher, A. Wagler, A construction for non-rank facets of stable set polytopes of webs, European Journal of Combinatorics 27 (2006) 1172–1185; A. Pêcher, A. Wagler, Almost all webs are not rank-perfect, Mathematical Programming Series B 105 (2006) 311–328] and Wagler [A. Wagler, Relaxing perfectness: Which graphs are ‘Almost’ perfect?, in: M. Groetschel (Ed.), The Sharpest Cut, Impact of Manfred Padberg and his work, in: SIAM/MPS Series on Optimization, vol. 4, Philadelphia, 2004; A. Wagler, Antiwebs are rank-perfect, 4OR 2 (2004) 149–152].     

Nonlinear Dunkl-wave equations

In this article, we introduce a class of nonlinear wave equations associated with the Dunkl operators, we study local and global well-posedness. Next, we establish the linearization of bounded energy solutions in the spirit of Gérard [P. Gérard, Oscillations and concentration effects in semilinear dispersive wave equations, J. Funct. Anal. 141 (1996), pp. 60–98]. The proof uses Strichartz-type inequalities and the energy estimate.     

On fractional Poincaré inequalities

We show that fractional (p, p)-Poincaré inequalities and even fractional Sobolev-Poincaré inequalities hold for bounded John domains, and especially for bounded Lipschitz domains. We also prove sharp fractional (1,p)-Poincaré inequalities for s-John domains.     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

Convex Sobolev inequalities and spectral gap

This Note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux, and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities including in the limit case corresponding to the logarithmic Sobolev inequalities. To cite this article: J.-P. Bartier, J. Dolbeault, C. R. Acad. Sci. Paris, Ser. I 342 (2006).     

An Alexandroff–Bakelman–Pucci estimate on Riemannian manifolds

In Cabré (1997) [2], Cabré established an Alexandroff–Bakelman–Pucci (ABP) estimate on Riemannian manifolds with non-negative sectional curvatures and applied it to establish the Krylov–Safonov Harnack inequality on manifolds with non-negative sectional curvatures. In the present paper, we generalize the results of [2]. We obtain an ABP estimate on manifolds with Ricci curvatures bounded from below and apply this estimate to prove the Krylov–Safonov Harnack inequality on manifolds with sectional curvatures bounded from below. We also use this ABP estimate to study Minkowski-type inequalities.     

Differential mixed variational inequalities in finite dimensional spaces

In this paper, we introduce and study a class of differential mixed variational inequalities in finite dimensional Euclidean spaces. Under various conditions, we obtain linear growth and bounded linear growth of the solution set for the mixed variational inequalities. Moreover, we present some conclusions which enrich the literature on the mixed variational inequalities and generalize the corresponding results of [4]. In particular we prove existence theorems for weak solutions of a differential mixed variational inequality in the weak sense of Carathéodory by using a result on differential inclusions involving an upper semicontinuous set-valued map with closed convex values. Also by employing the results from differential inclusions we establish a convergence result on Euler time-dependent procedure for solving initial-value differential mixed variational inequalities.     

Derivative Formula and Harnack Inequality for SDEs Driven by Lévy Processes

By using lower bound conditions of the Lévy measure, derivative formulae and Harnack inequalities are derived for linear stochastic differential equations driven by Lévy processes. As applications, explicit gradient estimates and heat kernel inequalities are presented. As byproduct, a new Girsanov theorem for Lévy processes is derived.     

Poincaré-type inequalities via stochastic integrals

Summary Martingales and stochastic integrals are applied to prove Poincaré-type inequalities involving probability distributions on the Euclidean space. These inequalities generalize and improve several results in the literature and are shown to yield weighted Poincaré inequalities on some special compact manifolds. This leads to a new method of calculating all the eigenvalues and eigenfunctions of the Laplacian on then-sphere. As a by-product the eigenvalues are shown to be related to the moments of a probability distribution.     

A note on a poincaré type inequality for solutions to subelliptic equations

We prove Poincaré type inequalities for solutions to certain classes of quasilinear subelliptic equations, including the well-known p-Sublaplacian. A notable feature in these inequalities is to replace the usual ?_B, the average of ? over a metric ball B, by ?(x₀) for x₀ ?B. Result of this kind was considered earlier by Ziemer [18] in the classic case. We mention that our endpoint result, even in the classic case, is not obtainable through the compactness argument.     

Interpolation, correlation identities, and inequalities for infinitely divisible variables

We present an interpolation formula for the expectation of functions of infinitely divisible (i.d.) variables. This is then applied to study the association problem for i.d. vectors and to present new covariance expansions and correlation inequalities. Acknowledgements and Notes. The research of C. Houdré was supported in part by an NSF Mathematical Sciences Post-Doctoral Fellowship and by an NSF-NATO Postdoctoral Fellowship and by the NSF grant No. DMS-98032039. This research was completed while V. Pérez-Abreu was visiting the Georgia Institute of Technology.     

On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems,including inequalities for log concave functions,and with an application to the diffusion equation

We extend the Prékopa-Leindler theorem to other types of convex combinations of two positive functions and we strengthen the Prékopa-Leindler and Brunn-Minkowski theorems by introducing the notion of essential addition. Our proof of the Prékopa-Leindler theorem is simpler than the original one. We sharpen the inequality that the marginal of a log concave function is log concave, and we prove various moment inequalities for such functions. Finally, we use these results to derive inequalities for the fundamental solution of the diffusion equation with a convex potential.     

Functional Inequalities for Particle Systems on Polish Spaces

Various Poincaré–Sobolev type inequalities are studied for a reaction–diffusion model of particle systems on Polish spaces. The systems we consider consist of finite particles which are killed or produced at certain rates, while particles in the system move on the Polish space interacting with one another (i.e. diffusion). Thus, the corresponding Dirichlet form, which we call reaction–diffusion Dirichlet form, consists of two parts: the diffusion part induced by certain Markov processes on the product spaces Eⁿ (n≥1) which determine the motion of particles, and the reaction part induced by a Q-process on ℤ₊ and a sequence of reference probability measures, where the Q-process determines the variation of the number of particles and the reference measures describe the locations of newly produced particles. We prove that the validity of Poincaré and weak Poincaré inequalities are essentially due to the pure reaction part, i.e. either of these inequalities holds if and only if it holds for the pure reaction Dirichlet form, or equivalently, for the corresponding Q-process. But under a mild condition, stronger inequalities rely on both parts: the reaction–diffusion Dirichlet form satisfies a super Poincaré inequality (e.g., the log-Sobolev inequality) if and only if so do both the corresponding Q-process and the diffusion part. Explicit estimates of constants in the inequalities are derived. Finally, some specific examples are presented to illustrate the main results. Mathematics Subject Classifications (2000) 4FD0F, 60H10. Feng-Yu Wang: Supported in part by the DFG through the Forschergruppe “Spectral Analysis, Asymptotic Distributions and Stochastic Dynamics”, the BiBoS Research Centre, NNSFC(10121101), and RFDP(20040027009).     

Modulus and Poincaré Inequalities on Non-Self-Similar Sierpiński Carpets

A carpet is a metric space homeomorphic to the Sierpiński carpet. We characterize, within a certain class of examples, non-self-similar carpets supporting curve families of nontrivial modulus and supporting Poincaré inequalities. Our results yield new examples of compact doubling metric measure spaces supporting Poincaré inequalities: these examples have no manifold points, yet embed isometrically as subsets of Euclidean space.     

Fluctuations of eigenvalues and second order Poincaré inequalities

Linear statistics of eigenvalues in many familiar classes of random matrices are known to obey gaussian central limit theorems. The proofs of such results are usually rather difficult, involving hard computations specific to the model in question. In this article we attempt to formulate a unified technique for deriving such results via relatively soft arguments. In the process, we introduce a notion of ‘second order Poincaré inequalities’: just as ordinary Poincaré inequalities give variance bounds, second order Poincaré inequalities give central limit theorems. The proof of the main result employs Stein’s method of normal approximation. A number of examples are worked out, some of which are new. One of the new results is a CLT for the spectrum of gaussian Toeplitz matrices.