On a class of Rellich inequalities

We prove the Rellich and the improved Rellich inequalities that involve the distance function from a hypersurface of codimension k, under a certain geometric assumption. In case, the distance is taken from the boundary, that assumption is the convexity of the domain. We also discuss the best constant of these inequalities.     

On Rellich’s Inequalities in Euclidean Spaces

On domains of Euclidean spaces we consider inequalities for test functions and their Laplacians. We describe a family of domains having vanishing Rellich constants. For the Euclidean space of dimension 4 we present a new version of the Rellich inequality. In addition, we prove new one-dimensional Rellich-type integral inequalities for linear combinations of test functions and their derivatives of orders one and two.     

Rellich type inequalities in domains of the Euclidean space

For smooth functions supported in a domain of the Euclidean space we investigate two Rellich type inequalities with weights which are powers of the distance function. We prove that for an arbitrary plane domain there exist positive Rellich constants in these inequalities if and only if the boundary of the domain is a uniformly perfect set. Moreover, we obtain explicit estimates of constants in function of geometric domain characteristics. Also, we find sharp constants in these Rellich type inequalities for all non-convex domains of dimension d ≥ 2 provided that the domains satisfy the exterior sphere condition with certain restriction on the radius of spheres.     

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.      

Hardy and Rellich type inequalities on complete manifolds

We prove some Hardy and Rellich type inequalities on complete noncompact Riemannian manifolds supporting a weight function which is not very far from the distance function in the Euclidean space.     

SOME SHARP RELLICH TYPE INEQUALITIES ON NILPOTENT GROUPS AND APPLICATION

We prove some Rellich type inequalities for the sub-Laplacian on Carnot nilpotent groups.Using the same method,we obtain some analogous inequalities for the Heisenberg-Greiner operators.In most cases,the constants we obtained are optimal.     

Inequalities,assessment and computer algebra

The goal of this paper is to examine single variable real inequalities that arise as tutorial problems and to examine the extent to which current computer algebra systems (CAS) can (1) automatically solve such problems and (2) determine whether students’ own answers to such problems are correct. We review how inequalities arise in contemporary curricula. We consider the formal mathematical processes by which such inequalities are solved, and we consider the notation and syntax through which solutions are expressed. We review the extent to which current CAS can accurately solve these inequalities, and the form given to the solutions by the designers of this software. Finally, we discuss the functionality needed to deal with students’ answers, i.e. to establish equivalence (or otherwise) of expressions representing unions of intervals. We find that while contemporary CAS accurately solve inequalities there is a wide variety of notation used.     

海森堡群上半空间内最佳Hardy不等式

证明了一类海森堡群上半空间内与次拉普拉斯算子相关的最佳Hardy不等式.作为应用,我们得到了相应的最佳Rellich型不等式.     

Conformal Killing Vector Fields and Rellich Type Identities on Riemannian Manifolds, II

We propose a general Noetherian approach to Rellich integral identities. Using this method we obtain a higher order Rellich type identity involving the polyharmonic operator on Riemannian manifolds admitting homothetic transformations. Then we prove a biharmonic Rellich identity in a more general context. We also establish a nonexistence result for semilinear systems involving biharmonic operators.     

Hardy–Rellich identities with Bessel pairs

Archiv der Mathematik - We prove an identity that implies the classical Rellich inequality as well as several improved versions of Rellich type inequalities. Moreover, our equality gives a simple...     

On Hardy–Rellich type inequalities in RN

In this note we introduce a new class of Hardy–Rellich type inequalities and explicitly obtain their corresponding sharp constants. Our approach suggests definitions of new Sobolev spaces and embedding results.     

On a class of weighted anisotropic Sobolev inequalities

In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities where different derivatives have different weight functions. These inequalities are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider Sobolev inequalities on finite cylinders, the weight being a power of the distance function from the top or the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is a power of the distance function from a higher codimension part of the boundary.     

Orlicz-Sobolev inequalities for sub-Gaussian measures and ergodicity of Markov semi-groups

We study coercive inequalities in Orlicz spaces associated to the probability measures on finite- and infinite-dimensional spaces which tails decay slower than the Gaussian ones. We provide necessary and sufficient criteria for such inequalities to hold and discuss relations between various classes of inequalities.     

Functional inequalities for transient birth-death processes and their applications

We give a new diagram about uniform decay, empty essential spectrum and various functional inequalities, including Poincaré inequalities, super- and weak-Poincaré inequalities, for transient birth-death processes. This diagram is completely opposite to that in ergodic situation, and substantially points out the difference between transient birth-death processes and recurrent ones. The criterion for the empty essential spectrum is achieved. Some matching sufficient and necessary conditions for weak-Poincaré inequalities and super-Poincaré inequalities are also presented.     

Mixing mixed-integer inequalities

Mixed-integer rounding (MIR) inequalities play a central role in the development of strong cutting planes for mixed-integer programs. In this paper, we investigate how known MIR inequalities can be combined in order to generate new strong valid inequalities.?Given a mixed-integer region S and a collection of valid “base” mixed-integer inequalities, we develop a procedure for generating new valid inequalities for S. The starting point of our procedure is to consider the MIR inequalities related with the base inequalities. For any subset of these MIR inequalities, we generate two new inequalities by combining or “mixing” them. We show that the new inequalities are strong in the sense that they fully describe the convex hull of a special mixed-integer region associated with the base inequalities.?We discuss how the mixing procedure can be used to obtain new classes of strong valid inequalities for various mixed-integer programming problems. In particular, we present examples for production planning, capacitated facility location, capacitated network design, and multiple knapsack problems. We also present preliminary computational results using the mixing procedure to tighten the formulation of some difficult integer programs. Finally we study some extensions of this mixing procedure. Received: April 1998 / Accepted: January 2001?Published online April 12, 2001     

Best constants for Sobolev inequalities for higher order fractional derivatives

We obtain sharp constants for Sobolev inequalities for higher order fractional derivatives. As an application, we give a new proof of a theorem of W. Beckner concerning conformally invariant higher-order differential operators on the sphere.     

Poincaré and Sobolev Inequalities for Vector Fields Satisfying Hrmander's Condition in Variable Exponent Sobolev Spaces

In this paper,we will establish Poincare inequalities in variable exponent non-isotropic Sobolev spaces.The crucial part is that we prove the boundedness of the fractional integral operator on variable exponent Lebesgue spaces on spaces of homogeneous type.We obtain the first order Poincare inequalities for vector fields satisfying Hrmander's condition in variable non-isotropic Sobolev spaces.We also set up the higher order Poincare inequalities with variable exponents on stratified Lie groups.Moreover,we get the Sobolev inequalities in variable exponent Sobolev spaces on whole stratified Lie groups.These inequalities are important and basic tools in studying nonlinear subelliptic PDEs with variable exponents such as the p(x)-subLaplacian.Our results are only stated and proved for vector fields satisfying Hrmander's condition,but they also hold for Grushin vector fields as well with obvious modifications.     

Hardy–Rellich inequalities with boundary remainder terms and applications

We prove a family of Hardy–Rellich inequalities with optimal constants and additional boundary terms. These inequalities are used to study the behavior of extremal solutions to biharmonic Gelfand-type equations under Steklov boundary conditions.     

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two‐weight Hardy, Hardy–Poincaré and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.