The shape of extremal functions for Poincaré-Sobolev-type inequalities in a ball

We study extremal functions for a family of Poincaré-Sobolev-type inequalities. These functions minimize, for subcritical or critical p?2, the quotient ‖∇u_‖2/‖up_‖ among all u∈H¹(B)?{0} with B_∫u=0. Here B is the unit ball in RN. We show that the minimizers are axially symmetric with respect to a line passing through the origin. We also show that they are strictly monotone in the direction of this line. In particular, they take their maximum and minimum precisely at two antipodal points on the boundary of B. We also prove that, for p close to 2, minimizers are antisymmetric with respect to the hyperplane through the origin perpendicular to the symmetry axis, and that, once the symmetry axis is fixed, they are unique (up to multiplication by a constant). In space dimension two, we prove that minimizers are not antisymmetric for large p.     

On the best constant of weighted Poincaré inequalities

We compute the optimal constant for some weighted Poincaré inequalities obtained by Fausto Ferrari and Enrico Valdinoci in [F. Ferrari, E. Valdinoci, Some weighted Poincaré inequalities, Indiana Univ. Math. J. 58 (4) (2009) 1619-1637].     

Lyapunov conditions for Super Poincaré inequalities

We show how to use Lyapunov functions to obtain functional inequalities which are stronger than Poincaré inequality (for instance logarithmic Sobolev or F-Sobolev). The case of Poincaré and weak Poincaré inequalities was studied in [D. Bakry, P. Cattiaux, A. Guillin, Rate of convergence for ergodic continuous Markov processes: Lyapunov versus Poincaré, J. Funct. Anal. 254 (3) (2008) 727-759. Available on Mathematics arXiv:math.PR/0703355, 2007]. This approach allows us to recover and extend in a unified way some known criteria in the euclidean case (Bakry and Emery, Wang, Kusuoka and Stroock, …).     

Sub-elliptic global high order Poincaré inequalities in stratified Lie groups and applications

Sharp Poincaré inequalities on balls or chain type bounded domains have been extensively studied both in classical Euclidean space and Carnot-Carathéodory spaces associated with sub-elliptic vector fields (e.g., vector fields satisfying Hörmander's condition). In this paper, we investigate the validity of sharp global Poincaré inequalities of both first order and higher order on the entire nilpotent stratified Lie groups or on unbounded extension domains in such groups. We will show that simultaneous sharp global Poincaré inequalities also hold and weighted versions of such results remain to be true. More precisely, let G be a nilpotent stratified Lie group and f be in the localized non-isotropic Sobolev space , where 1?p<Q/m and Q is the homogeneous dimension of the Lie group G. Suppose that the mth sub-elliptic derivatives of f is globally Lp integrable; i.e., is finite (but assume that lower order sub-elliptic derivatives are only locally Lp integrable). We denote the space of such functions as Bm,p(G). We prove a high order Poincaré inequality for f minus a polynomial of order m−1 over the entire space G or unbounded extension domains. As applications, we will prove a density theorem stating that smooth functions with compact support are dense in Bm,p(G) modulus a finite-dimensional subspace.     

Self-improving properties of inequalities of Poincaré type on measure spaces and applications

We show that the self-improving nature of Poincaré estimates persists for domains in rather general measure spaces. We consider both weak type and strong type inequalities, extending techniques of B. Franchi, C. Pérez and R. Wheeden. As an application in spaces of homogeneous type, we derive global Poincaré estimates for a class of domains with rough boundaries that we call ?-John domains, and we show that such domains have the requisite properties. This class includes John (or Boman) domains as well as s-John domains. Further applications appear in a companion paper.     

On Friedrichs-Poincaré-type inequalities

Friedrichs- and Poincaré-type inequalities are important and widely used in the area of partial differential equations and numerical analysis. Most of their proofs appearing in references are the argument of reduction to absurdity. In this paper, we give direct proofs of Friedrichs-type inequalities in H¹(Ω) and Poincaré-type inequalities in some subspaces of W1,p(Ω). The dependencies of the inequality coefficients on the domain Ω and some sub-domains are illustrated explicitly.     

Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy-entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré-type inequalities with weights are derived. Optimal constants are compared to the usual Poincaré constant.     

A CR Poincaré inequality on the complex sphere

The main purpose of this paper is to prove a CR Poincaré inequality with sharp exponent on the sphere in complex space. We use the complex tangential gradient on the sphere instead of the usual Laplace-Beltrami gradient on the sphere.     

Weak curvature conditions and functional inequalities

We give sufficient conditions for a measured length space (X,d,ν) to admit local and global Poincaré inequalities, along with a Sobolev inequality. We first introduce a condition DM on (X,d,ν), defined in terms of transport of measures. We show that DM, together with a doubling condition on ν, implies a scale-invariant local Poincaré inequality. We show that if (X,d,ν) has nonnegative N-Ricci curvature and has unique minimizing geodesics between almost all pairs of points then it satisfies DM, with constant N². The condition DM is preserved by measured Gromov-Hausdorff limits. We then prove a Sobolev inequality for measured length spaces with N-Ricci curvature bounded below by K>0. Finally we derive a sharp global Poincaré inequality.     

Improved Sobolev inequalities and Muckenhoupt weights on stratified Lie groups

We study in this article the improved Sobolev inequalities with Muckenhoupt weights within the framework of stratified Lie groups. This family of inequalities estimate the Lq norm of a function by the geometric mean of two norms corresponding to Sobolev spaces and Besov spaces . When the value p which characterizes Sobolev space is strictly larger than 1, the required result is well known in Rn and is classically obtained by a Littlewood-Paley dyadic blocks manipulation. For these inequalities we will develop here another totally different technique. When p=1, these two techniques are not available anymore and following M. Ledoux (2003) [12], in Rn, we will treat here the critical case p=1 for general stratified Lie groups in a weighted functional space setting. Finally, we will go a step further with a new generalization of improved Sobolev inequalities using weak-type Sobolev spaces.     

Poincaré and Opial inequalities for vector-valued convolution products

Various L^p$L^p$ form Poincaré and Opial inequalities are given for vector-valued convolution products. We apply our results to infinitesimal generators of _C0$C_0$-semigroups and cosine functions. Typical examples of these operators are differential operators in Lebesgue spaces.     

From concentration to logarithmic Sobolev and Poincaré inequalities

We give a new proof of the fact that Gaussian concentration implies the logarithmic Sobolev inequality when the curvature is bounded from below, and also that exponential concentration implies Poincaré inequality under null curvature condition. Our proof holds on non-smooth structures, such as length spaces, and provides a universal control of the constants. We also give a new proof of the equivalence between dimension free Gaussian concentration and Talagrand's transport inequality.     

Improved Poincaré inequalities

Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to the quadratic singularity without violating the inequality, and even a whole asymptotic expansion can be built, with optimal constants for each term. This phenomenon has not been much studied for other inequalities. Our purpose is to prove that it also holds for the gaussian Poincaré inequality. The method is based on a recursion formula, which allows to identify the optimal constants in the asymptotic expansion, order by order. We also apply the same strategy to a family of Hardy–Poincaré inequalities which interpolate between Hardy and gaussian Poincaré inequalities.     

Eigenvalue gaps for the Cauchy process and a Poincaré inequality

A connection between the semigroup of the Cauchy process killed upon exiting a domain D and a mixed boundary value problem for the Laplacian in one dimension higher known as the mixed Steklov problem, was established in [R. Bañuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 211 (2004) 355-423]. From this, a variational characterization for the eigenvalues λn, n?1, of the Cauchy process in D was obtained. In this paper we obtain a variational characterization of the difference between λn and λ₁. We study bounded convex domains which are symmetric with respect to one of the coordinate axis and obtain lower bound estimates for λ_∗−λ₁ where λ_∗ is the eigenvalue corresponding to the “first” antisymmetric eigenfunction for D. The proof is based on a variational characterization of λ_∗−λ₁ and on a weighted Poincaré-type inequality. The Poincaré inequality is valid for all α symmetric stable processes, 0<α?2, and any other process obtained from Brownian motion by subordination. We also prove upper bound estimates for the spectral gap λ₂−λ₁ in bounded convex domains.     

Asymmetric Poincaré inequalities and solvability of capillarity problems

In this paper we first establish an asymmetric version of the Poincaré inequality in the space of bounded variation functions, and then, basically relying on this result, we discuss the existence, the non-existence and the multiplicity of bounded variation solutions of a class of capillarity problems with asymmetric perturbations.     

Poincaré inequalities, uniform domains and extension properties for Newton-Sobolev functions in metric spaces

In the setting of metric measure spaces equipped with a doubling measure supporting a weak p-Poincaré inequality with 1?p<∞, we show that any uniform domain Ω is an extension domain for the Newtonian space N1,p(Ω) and that Ω, together with the metric and the measure inherited from X, supports a weak p-Poincaré inequality. For p>1, we obtain a near characterization of N1,p-extension domains with local estimates for the extension operator.     

Mixed exterior Laplace's problem

In [C. Amrouche, V. Girault, J. Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator. An approach in weighted Sobolev spaces, J. Math. Pures Appl. 76 (1997) 55-81], authors study Dirichlet and Neumann problems for the Laplace operator in exterior domains of Rn. This paper extends this study to the resolution of a mixed exterior Laplace's problem. Here, we give existence, unicity and regularity results in Lp's theory with 1<p<∞, in weighted Sobolev spaces.     

Second order Poincaré inequalities and CLTs on Wiener space

We prove infinite-dimensional second order Poincaré inequalities on Wiener space, thus closing a circle of ideas linking limit theorems for functionals of Gaussian fields, Stein's method and Malliavin calculus. We provide two applications: (i) to a new “second order” characterization of CLTs on a fixed Wiener chaos, and (ii) to linear functionals of Gaussian-subordinated fields.     

Poincaré's inequality and Sobolev spaces with monomial weights

In this paper, we use a weighted version of Poincaré's inequality to study density and extension properties of weighted Sobolev spaces over some open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$. Additionally, we study the specific case of monomial weights $w(x_1,\text{…},x_N)={\prod }_{i=1}^N{\left|x_i\right|}^{a_i},a_i\ge 0$, showing the validity of a weighted Poincaré inequality together with some embedding properties of the associated weighed Sobolev spaces.