Search

We establish improved Hardy inequalities for the magnetic p -Laplacian due to adding nontrivial magnetic fields. We also prove that for Aharonov–Bohm magnetic fields the sharp constant in the Hardy inequality becomes strictly larger than in the case of a magnetic-free p -Laplacian. We also post some remarks with open problems. [ABSTRACT FROM AUTHOR]

In this paper, some sharp Sobolev inequalities on $ BV({\mathbb{R}}^n) $, the space of functions of bounded variation on $ {\mathbb{R}}^n $, $ n\geq 2 $, are deduced through the $ L_p $ Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on $ BV({\mathbb{R}}^n) $.

The carré du champ method is a powerful technique for proving interpolation inequalities with explicit constants in presence of a non-trivial metric on a manifold. The method applies to some classical Gagliardo-Nirenberg-Sobolev inequalities on the sphere, with optimal constants. Very nonlinear regimes close to the critical Sobolev exponent can be covered using nonlinear parabolic flows of porous medium or fast diffusion type. Considering power law weights is a natural question in relation with symmetry breaking issues for Caffarelli-Kohn-Nirenberg inequalities, but regularity estimates for a complete justification of the computation are missing. We provide the first example of a complete parabolic proof based on a nonlinear flow by regularizing the singularity induced by the weight. Our result is established in the simplified framework of a diffusion built on the ultraspherical operator, which amounts to reduce the problem to functions on the sphere with simple symmetry properties. [ABSTRACT FROM AUTHOR]

In this paper, some sharp Sobolev inequalities on BV(Rn), the space of functions of bounded variation on Rn, n ≥ 2, are deduced through the Lp Brunn-Minkowski theory. We will prove that these inequalities can all imply the sharp Sobolev inequality on BV(Rn) [ABSTRACT FROM AUTHOR]

In this survey we give a compact presentation of well-known functional inequalities of Hardy and Rellich type in the L2 setting. In addition, we give some insights into the proofs by using basic tools with emphasis on the particularities of a more general approach which is the method of super-solutions. [ABSTRACT FROM AUTHOR]

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3. [ABSTRACT FROM AUTHOR]

In this paper, we study derivatives of powers of Euclidean norm. We prove their Hölder continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most two times suboptimal for the even ones. In the particular case of integer powers, when the Hölder continuity transforms into the Lipschitz continuity, we improve this result and obtain the optimal constants. [ABSTRACT FROM AUTHOR]

We derive an improved Poincaré inequality in connection with the Babuška{Aziz and Friedrichs{Velte inequalities for differential forms by estimating the domain specific optimal constants in the respective inequalities with each other provided the domain supports the Hardy inequality. We also derive upper estimates for the constants of a star-shaped domain by generalizing the known Horgan{Payne type estimates for planar and spatial domains to higher dimensional ones. [ABSTRACT FROM AUTHOR]

Abstract Let d ≥ 2 be an integer and let 2 d / (d − 1) < q ≤ ∞. In this paper we investigate the sharp form of the mixed norm Fourier extension inequality ‖ f σ ˆ ‖ L rad q L ang 2 (R d) ≤ C d , q ‖ f ‖ L 2 (S d − 1 , d σ) , established by L. Vega in 1988. Letting A d ⊂ (2 d / (d − 1) , ∞ ] be the set of exponents for which the constant functions on S d − 1 are the unique extremizers of this inequality, we show that: (i) A d contains the even integers and ∞; (ii) A d is an open set in the extended topology; (iii) A d contains a neighborhood of infinity (q 0 (d) , ∞ ] with q 0 (d) ≤ (1 2 + o (1)) d log ⁡ d. In low dimensions we show that q 0 (2) ≤ 6.76 ; q 0 (3) ≤ 5.45 ; q 0 (4) ≤ 5.53 ; q 0 (5) ≤ 6.07. In particular, this breaks for the first time the even exponent barrier in sharp Fourier restriction theory. The crux of the matter in our approach is to establish a hierarchy between certain weighted norms of Bessel functions, a nontrivial question of independent interest within the theory of special functions. [ABSTRACT FROM AUTHOR]

The Bohnenblust-Hille inequality for m-linear forms was proven in 1931 as a generalization of the famous 4/3-Littlewood inequality. The optimal constants (or at least their asymptotic behavior as m grows) is unknown, but significant for applications. A recent result, obtained by Cavalcante, Teixeira and Pellegrino, provides a kind of algorithm, composed by finitely many elementary steps, giving as the final outcome the optimal truncated Bohnenblust-Hille constants of any order. But the procedure of Cavalcante et al. has a fairly large number of calculations and computer assistance cannot be avoided. In this paper we present a computational solution to the problem, using the Wolfram Language. We also use this approach to investigate a conjecture raised by Pellegrino and Teixeira, asserting that for all m ∈ ℕ and to reveal interesting unknown facts about the geometry of . [ABSTRACT FROM AUTHOR]

For an appropriate class of convex functions φ, we study the Fourier extension operator on the surface {(y, |y|² + φ(y)) : y ∈ R²} equipped with projection measure. For the corresponding extension inequality, we compute optimal constants and prove that extremizers do not exist. The main tool is a new comparison principle for convolutions of certain singular measures that holds in all dimensions. Using tools of concentration-compactness flavor, we further investigate the behavior of general extremizing sequences. Our work is directly related to the study of extremizers and optimal constants for Strichartz estimates of certain higher order Schrödinger equations. In particular, we resolve a dichotomy from the recent literature concerning the existence of extremizers for a family of fourth order Schrödinger equations and compute the corresponding operator norms exactly where only lower bounds were previously known. [ABSTRACT FROM AUTHOR]

We study some Hardy-type inequalities involving a general norm in ℝn and an anisotropic distance function to the boundary. The case of the optimality of the constants is also addressed. [ABSTRACT FROM AUTHOR]

Recently, several authors have considered the problem of determining optimal norm inequalities for discrete Hardy-type operators (like Cesàro or Copson). In this work, we obtain sharp bounds for the norms of the difference of the Cesàro operator with either the identity or the shift, when they are restricted to the cone of decreasing sequences in ℓ p (which is closely related to the previously mentioned estimates). Finally, we also address the case of weighted inequalities and find an interesting contrast between the norms of these two difference operators. [ABSTRACT FROM AUTHOR]

In this paper we provide a family of inequalities, extending a recent result due to Albuquerque et al. [ABSTRACT FROM AUTHOR]

In this paper, we study the optimization problem in a suitable functional space . Here, is the multi-singular potential given by and all the singular poles , , arise either in the interior or at the boundary of a smooth open domain , with or , respectively. For a bounded domain containing all the singularities in the interior, we prove that when and when (it is known from [C. Cazacu and E. Zuazua, Improved multipolar hardy inequalities, in Studies in Phase Space Analysis with Application to PDEs, Progress in Nonlinear Differential Equations and Their Applications, Vol. 84 (Birkhäuser, New York, 2013), pp. 37-52] that . In the situation when all the poles are located on the boundary, we show that if is either a ball, the exterior of a ball or a half-space. Our results do not depend on the distances between the poles. In addition, in the case of boundary singularities we obtain that is attained in when is a ball and . Besides, is attained in when is the exterior of a ball with and whereas in the case of a half-space is attained in when . We also analyze the critical constants in the so-called weak Hardy inequality which characterizes the range of ensuring the existence of a lower bound for the spectrum of the Schrödinger operator . In the context of both interior and boundary singularities, we show that the critical constants in the weak Hardy inequality are and , respectively. [ABSTRACT FROM AUTHOR]

The well-known Gevrey classes consist of C ∞ -functions on R d whose derivatives satisfy certain growth conditions. For periodic functions these conditions can be expressed in terms of Fourier coefficients. Motivated by this observation, we introduce Gevrey spaces on the d -dimensional torus and study approximation numbers of their embeddings into L 2. Our special emphasis is on the dependence on the dimension of the underlying domain, which is an important aspect in the numerical treatment of high-dimensional problems. In particular, we determine the exact rate of decay of the approximation numbers, together with optimal asymptotic constants, and establish preasymptotic estimates. Finally we translate our findings into the language of tractability notions from information-based complexity. [ABSTRACT FROM AUTHOR]

For the solution of the free Schrödinger equation, we obtain the optimal constants and characterise extremisers for forward and reverse smoothing estimates which are global in space and time, contain a homogeneous and radial weight in the space variable, and incorporate a certain angular regularity. This will follow from a more general result which permits analogous sharp forward and reverse smoothing estimates and a characterisation of extremisers for the solution of the free Klein-Gordon and wave equations. The nature of extremisers is shown to be sensitive to both the dimension and the size of the smoothing index relative to the dimension. Furthermore, in four spatial dimensions and certain special values of the smoothing index, we obtain an exact identity for each of these evolution equations. [ABSTRACT FROM AUTHOR]

In this paper, we generalize the Pecaric work on Montgomery's identity via an arbitrary weight function, which no longer needs to be a probability density function, and apply it to derive some generalized Chebyshev type inequalities for any absolutely continuous function. [ABSTRACT FROM AUTHOR]

Abstract: We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolevʼs inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles. [Copyright &y& Elsevier]

Abstract: We obtain sharp estimates of the Hardy–Vitali type total -variation of a function of two variables in terms of its mixed modulus of continuity in . [Copyright &y& Elsevier]

Abstract: In this paper we present two-sided Chernoff-type inequalities for the error of the best approximation of a smooth d-variate function by polynomials of total degree less than k in the -norm. It is supposed that the d-variate weight function w has one-dimensional classical component weights, which satisfy Pearson differential equations. Similarly as in the univariate case, the leading coefficients of the multivariate classical polynomials, orthonormal with respect to the weight function w, play an important role in the presented estimates. [Copyright &y& Elsevier]

An inequality of Hardy type is established for quadratic forms involving Dirac operator and a weight r− b for functions in $${\mathbb{R}^n}$$. The exact Hardy constant c b = c b( n) is found and generalized minimizers are given. The constant c b vanishes on a countable set of b, which extends the known case n = 2, b = 0 which corresponds to the trivial Hardy inequality in $${\mathbb{R}^2}$$. Analogous inequalities are proved in the case c b = 0 under constraints and, with error terms, for a bounded domain. [ABSTRACT FROM AUTHOR]

Abstract: We use a variational approach to find the best constants for certain Gagliardo–Nirenberg inequalities on the real line. To show the existence of a minimizer, we use the method of concentration–compactness. [Copyright &y& Elsevier]

We consider the two dimensional gravitational Vlasov–Poisson system. Using variational methods, we prove the existence of stationary solutions of minimal energy under a Casimir type constraint. The method also provides a stability criterion of these solutions for the evolution problem. [ABSTRACT FROM AUTHOR]

Abstract: This paper is devoted to inequalities of Lieb–Thirring type. Let V be a nonnegative potential such that the corresponding Schrödinger operator has an unbounded sequence of eigenvalues . We prove that there exists a positive constant , such that, if , then and determine the optimal value of . Such an inequality is interesting for studying the stability of mixed states with occupation numbers. We show how the infimum of on all possible potentials V, which is a lower bound for , corresponds to the optimal constant of a subfamily of Gagliardo–Nirenberg inequalities. This explains how (∗) is related to the usual Lieb–Thirring inequality and why all Lieb–Thirring type inequalities can be seen as generalizations of the Gagliardo–Nirenberg inequalities for systems of functions with occupation numbers taken into account. We also state a more general inequality of Lieb–Thirring type where F and G are appropriately related. As a special case corresponding to , (∗∗) is equivalent to an optimal Euclidean logarithmic Sobolev inequality where , is any nonnegative sequence of occupation numbers and is any sequence of orthonormal functions. [Copyright &y& Elsevier]

Abstract: We give improved results on the existence of positive solutions for a nonlinear differential equation with nonlocal boundary conditions, that arise as a model of a thermostat. We obtain some optimal criteria for the existence of one positive solution which involve the principal eigenvalue of a related linear operator. We also determine optimal values of some other constants that are useful in obtaining existence of multiple positive solutions. [Copyright &y& Elsevier]

Abstract: We prove an optimal logarithmic Hardy–Littlewood–Sobolev inequality for systems on compact -dimensional Riemannian manifolds, for any . We show that a special case of the inequality, involving only two functions, implies the general case by using an argument from the theory of linear programing. [Copyright &y& Elsevier]

We prove some sharp Hardy-type inequalities related to the Dirac operator by elementary, direct methods. Some of these inequalities have been obtained previously using spectral information about the Dirac–Coulomb operator. Our results are stated under optimal conditions on the asymptotics of the potentials near zero and near infinity. [Copyright &y& Elsevier]

We prove a general optimal Lp-Euclidean logarithmic Sobolev inequality by using Pre´kopa–Leindler inequality and a special Hamilton–Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.). [Copyright &y& Elsevier]

We prove an optimal logarithmic Sobolev inequality in W1,p(Rd). Explicit minimizers are given. This result is connected with best constants of a special class of Gagliardo–Nirenberg-type inequalities. [Copyright &y& Elsevier]

Let Ω be a domain in R d and d Γ the Euclidean distance to the boundary Γ. We investigate whether the weighted Hardy inequality ‖ d Γ δ / 2 − 1 φ ‖ 2 ≤ a δ ‖ d Γ δ / 2 (∇ φ) ‖ 2 is valid, with δ ≥ 0 and a δ > 0 , for all φ ∈ C c 1 (Γ r) and all small r > 0 where Γ r = { x ∈ Ω : d Γ (x) < r }. First we prove that if δ ∈ [ 0 , 2 〉 then the inequality is equivalent to the weighted version of Davies' weak Hardy inequality on Ω with equality of the corresponding optimal constants. Secondly, we establish that if Ω is a uniform domain with a locally uniform Ahlfors regular boundary then the inequality is satisfied for all δ ≥ 0 , and all small r , with the exception of the value δ = 2 − (d − d H) where d H is the Hausdorff dimension of Γ. Moreover, the optimal constant a δ (Γ) satisfies a δ (Γ) ≥ 2 / | (d − d H) + δ − 2 |. Thirdly, if Ω is a C 1 , 1 -domain or a convex domain a δ (Γ) = 2 / | δ − 1 | for all δ ≥ 0 with δ ≠ 1. The same conclusion is correct if Ω is the complement of a convex domain and δ > 1 but if δ ∈ [ 0 , 1 〉 then a δ (Γ) can be strictly larger than 2 / | δ − 1 |. Finally we use these results to establish self-adjointness criteria for degenerate elliptic diffusion operators. [ABSTRACT FROM AUTHOR]

The main results of this paper state optimal constants for estimates of so-called successive minima in two dimensions under a constraint on the denominator. While these inequalities are known for every dimension, best possible constants within these estimates are, of course, notknown for any dimension larger than one and remain unknown for all dimensions larger than two. [ABSTRACT FROM AUTHOR]

Copyright of Meccanica is the property of Springer Nature and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)