A global Lipschitz continuity result for a domain-dependent Neumann eigenvalue problem for the Laplace operator

Let Ω be an open connected subset of Rⁿ of finite measure for which the Poincaré-Wirtinger inequality holds. We consider the Neumann eigenvalue problem for the Laplace operator in the open subset φ(Ω) of Rⁿ, where φ is a locally Lipschitz continuous homeomorphism of Ω onto φ(Ω). Then, we show Lipschitz-type inequalities for the reciprocals of the eigenvalues delivered by the Rayleigh quotient. Then, we further assume that the imbedding of the Sobolev space W^1,2(Ω) into the space L²(Ω) is compact, and we prove the same type of inequalities for the projections onto the eigenspaces upon variation of φ.     

Capacitary Criteria for Poincaré-Type Inequalities

The Poincaré-type inequality is a unification of various inequalities including the F-Sobolev inequalities, Sobolev-type inequalities, logarithmic Sobolev inequalities, and so on. The aim of this paper is to deduce some unified upper and lower bounds of the optimal constants in Poincaré-type inequalities for a large class of normed linear (Banach, Orlicz) spaces in terms of capacity. The lower and upper bounds differ only by a multiplicative constant, and so the capacitary criteria for the inequalities are also established. Both the transient and the ergodic cases are treated. Besides, the explicit lower and upper estimates in dimension one are computed. Mathematics Subject Classifications (2000) 60J55, 31C25, 60J35, 47D07.Research supported in part NSFC (No. 10121101) and 973 Project.     

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 note on Hardy’s inequalities with boundary singularities

Let Ω be a smooth bounded domain in R^N with N≥1. In this paper we study the Hardy-Poincaré inequalities with weight function singular at the boundary of Ω. In particular we give sufficient conditions so that the best constant is achieved.     

Improved Poincaré inequalities with weights

In this paper we prove that if Ω∈Rn is a bounded John domain, the following weighted Poincaré-type inequality holds:

     

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.     

Mass transportation proofs of free functional inequalities, and free Poincaré inequalities

This work is devoted to direct mass transportation proofs of families of functional inequalities in the context of one-dimensional free probability, avoiding random matrix approximation. The inequalities include the free form of the transportation, Log-Sobolev, HWI interpolation and Brunn-Minkowski inequalities for strictly convex potentials. Sharp constants and some extended versions are put forward. The paper also addresses two versions of free Poincaré inequalities and their interpretation in terms of spectral properties of Jacobi operators. The last part establishes the corresponding inequalities for measures on R₊ with the reference example of the Marcenko-Pastur distribution.     

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.     

Hardy and Rellich-Type Inequalities for Metrics Defined by Vector Fields

Let {X _i }, i=1,...,m be a system of locally Lipschitz vector fields on DR ⁿ , such that the corresponding intrinsic metric is well-defined and continuous w.r.t. the Euclidean topology. Suppose that the Lebesgue measure is doubling w.r.t. the intrinsic balls, that a scaled L¹ Poincaré inequality holds for the vector fields at hand (thus including the case of Hörmander vector fields) and that the local homogeneous dimension near a point x ₀ is sufficiently large. Then weighted Sobolev–Poincaré inequalities with weights given by power of (,x ₀) hold; as particular cases, they yield non-local analogues of both Hardy and Sobolev–Okikiolu inequalities. A general argument which shows how to deduce Rellich-type inequalities from Hardy inequalities is then given: this yields new Rellich inequalities on manifolds and even in the uniformly elliptic case. Finally, applications of Sobolev–Okikiolu inequalities to heat kernel estimates for degenerate subelliptic operators and to criteria for the absence of bound states for Schrödinger operators H=–L+V are given.     

Korn inequality on irregular domains

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.     

Functional Inequalities and Spectrum Estimates: The Infinite Measure Case

As a continuation of [13] where a Poincaré-type inequality was introduced to study the essential spectrum on the L²-space of a probability measure, this paper provides a modification of this inequality so that the infimum of the essential spectrum is well described even if the reference measure is infinite. High-order eigenvalues as well as the corresponding semigroup are estimated by using this new inequality. Criteria of the inequality and estimates of the inequality constants are presented. Finally, some concrete examples are considered to illustrate the main results. In particular, estimates of high-order eigenvalues obtained in this paper are sharp as checked by two examples on the Euclidean space.     

On discrete inequalities of the Poincaré type

The aim of the present paper is to establish some new discrete inequalities of the Poincaré type involving functions ofn independent variables and their first order forward differences. The proofs given here are quite elementary and our results provide new estimates on this type of discrete inequalities.     

Estimates for derivatives of holomorphic functions in a hyperbolic domain

Let f(z) be a holomorphic function in a hyperbolic domain Ω. For 2?n?8, the sharp estimate of |f(n)(z)/f^′(z)| associated with the Poincaré density λΩ(z) and the radius of convexity ρΩc(z) at z∈Ω is established for f(z) univalent or convex in each Δc(z) and z∈Ω. The detailed equality condition of the estimate is given. Further application of the results to the Avkhadiev-Wirths conjecture is also discussed.     

Is an Orlicz–Poincaré inequality an open ended condition,and what does that mean?

In [16], Keith and Zhong prove that spaces admitting Poincaré inequalities also admit a priori stronger Poincaré inequalities. We use their technique, with slight adjustments, to obtain a similar result in the case of Orlicz–Poincaré inequalities. We give examples in the plane that show all hypotheses are required.     

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.     

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, …).     

Splitting theorem, Poincaré-Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll-Meyer splitting theorem and a shifting theorem for J∈C^2-0(E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré-Hopf theorem to J∈C¹(E,R) case. By combining the Poincaré-Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations.     

Some new and short proofs for a class of Caffarelli-Kohn-Nirenberg type inequalities

In this note we provide simple and short proofs for a class of inequalities of Caffarelli-Kohn-Nirenberg type with sharp constants. Our approach suggests some definitions of weighted Sobolev spaces and their embedding into weighted L² spaces. These may be useful in studying solvability of problems involving new singular PDEs.     

New Poincaré-type inequalities

We present some Poincaré-type inequalities for quadratic matrix fields with applications e.g. in gradient plasticity or fluid dynamics. In particular, applications to the pseudostress–velocity formulation of the stationary Stokes problem and to infinitesimal gradient plasticity are discussed.     

Global weighted inequalities for operators and harmonic forms on manifolds

In this paper, both the local and global weighted Sobolev-Poincaré imbedding inequalities and Poincaré inequalities for the composition T°G are established, where T is the homotopy operator and G is Green's operator applied to A-harmonic forms on manifolds.