A quantitative form of Faber–Krahn inequality

The Faber–Krahn inequality states that balls are the unique minimizers of the first eigenvalue of the p-Laplacian among all sets with fixed volume. In this paper we prove a sharp quantitative form of this inequality. This extends to the case \(p>1\) a recent result proved by Brasco et al. (Duke Math J 164:1777–1831, 2015) for the Laplacian.     

A Faber–Krahn Inequality for Solutions of Schrödinger’s Equation on Riemannian Manifolds

We consider a bounded open set with smooth boundary \(\Omega \subset M\) in a Riemannian manifold (M, g), and suppose that there exists a non-trivial function \(u\in C({\overline{\Omega }})\) solving the problem

$$\begin{aligned} -\Delta u=V(x)u, \,\, \text{ in }\,\,\Omega , \end{aligned}$$

in the distributional sense, with \(V\in L^\infty (\Omega )\), where \(u\equiv 0\) on \(\partial \Omega .\) We prove a sharp inequality involving \(||V||_{L^{\infty }(\Omega )}\) and the first eigenvalue of the Laplacian on geodesic balls in simply connected spaces with constant curvature, which slightly generalises the well-known Faber–Krahn isoperimetric inequality. Moreover, in a Riemannian manifold which is not necessarily simply connected, we obtain a lower bound for \(||V||_{L^{\infty }(\Omega )}\) in terms of its isoperimetric or Cheeger constant. As an application, we show that if \(\Omega \) is a domain on a m-dimensional minimal submanifold of \({\mathbb {R}}^n\) which lies in a ball of radius R, then

$$\begin{aligned} ||V||_{L^{\infty }(\Omega )}\ge \left( \frac{m}{2R}\right) ^{2}. \end{aligned}$$

     

The quantitative Faber–Krahn inequality for the Robin Laplacian

We prove a quantitative form of the Faber–Krahn inequality for the first eigenvalue of the Laplace operator with Robin boundary conditions. The asymmetry term involves the square power of the Fraenkel asymmetry, multiplied by a constant depending on the Robin parameter, the dimension of the space and the measure of the set.     

An alternative approach to the Faber–Krahn inequality for Robin problems

We give a simple proof of the Faber–Krahn inequality for the first eigenvalue of the p-Laplace operator with Robin boundary conditions. The techniques introduced allow to work with much less regular domains by using test function arguments. We substantially simplify earlier proofs, and establish the sharpness of the inequality for a larger class of domains at the same time.     

On the Exact Constant in the Jackson–Stechkin Inequality for the Uniform Metric

The classical Jackson–Stechkin inequality estimates the value of the best uniform approximation of a 2π-periodic function f by trigonometric polynomials of degree ≤n−1 in terms of its r-th modulus of smoothness ω _r (f,δ). It reads

A Class of Counterexamples to the Fefferman–Phong Inequality for Systems

Abstract

We give here a class of counterexamples to the Fefferman–Phong inequality for systems of pseudodifferential operators, which contains Brummelhuis’ one as a particular case. The main ingredient in the proof is the use of “localized operators” associated with the system, and Hörmander's example of a positive-semidefinite matrix whose Weyl quantization is not nonnegative. For the considered class, in the “isotropic” case, the Sharp Gårding inequality cannot be improved.     

A Generalization of the Logarithmic Gross–Sobolev Inequality

Doklady Mathematics - A sharp integral inequality is proved that is used to derive a Sobolev interpolation inequality. A generalization of the logarithmic Sobolev inequality is proposed based on...     

On the Hong–Krahn–Szego inequality for the p-Laplace operator

Given an open set Ω, we consider the problem of providing sharp lower bounds for λ ₂(Ω), i.e. its second Dirichlet eigenvalue of the p-Laplace operator. After presenting the nonlinear analogue of the Hong–Krahn–Szego inequality, asserting that the disjoint unions of two equal balls minimize λ ₂ among open sets of given measure, we improve this spectral inequality by means of a quantitative stability estimate. The extremal cases p = 1 and p = ∞ are considered as well.     

A Note for one of Inequality

In this paper, we find the largest contant C such that sin2 x xtanx > 2x2 cx5tan x 0相似文献   

Maximizers for the Stein–Tomas Inequality

We give a necessary and sufficient condition for the precompactness of all optimizing sequences for the Stein–Tomas inequality. In particular, if a well-known conjecture about the optimal constant in the Strichartz inequality is true, we obtain the existence of an optimizer in the Stein–Tomas inequality. Our result is valid in any dimension.     

Refinements of the Dubins–Savage Inequality

This paper sharpens the Dubins–Savage inequality for certain supermartingales whose conditional moment-generating functions are suitably bounded. In particular, sharper inequalities are derived for generalized gaussian, sub-normal and sub-Poisson sequences. A related inequality due to Khan is also refined.     

A Refinement of the Arithmetic-geometric Means Inequality

We suppose throughout that(1)m,n∈N,a_i,a_ij,q_j,p,x are all positive numbers;∑_j=1~n q_j=1,l≥1,λ>0,(i=1,2,…,m;j=1,2,…,n).     

Quasiunconditional Basis Property of the Faber–Schauder System

Ukrainian Mathematical Journal - We prove that, for any 0 < ?? < 1, there exists a measurable set E?? ? [0, 1], mes (E??) > 1...     

A Characterization of Some Mixed Volumes via the Brunn–Minkowski Inequality

We consider a functional $\mathcal{F}$ on the space of convex bodies in ? ⁿ of the form $$ {\mathcal{F}}(K)=\int_{\mathbb{S}^{n-1}} f(u) \mathrm{S}_{n-1}(K,du), $$ where $f\in C(\mathbb{S}^{n-1})$ is a given continuous function on the unit sphere of ? ⁿ , K is a convex body in ? ⁿ , n≥3, and S _n?1(K,?) is the area measure of K. We prove that $\mathcal{F}$ satisfies an inequality of Brunn–Minkowski type if and only if f is the support function of a convex body, i.e., $\mathcal{F}$ is a mixed volume. As a consequence, we obtain a characterization of translation invariant, continuous valuations which are homogeneous of degree n?1 and satisfy a Brunn–Minkowski type inequality.     

Ruelle''''s Inequality for the Entropy of Diffusion Processes

In this paper we prove the Ruelle's inequality for the entropy and Lyapunovexponents of diffusion processes, that is, suppose h_μ,λ(i)(x),m_i (x) are respectivelythe entropy, Lyapunov exponents and its multiplicity for the random diffeomorphismsarising from SDE:     

One Inequality of the Landau–Kolmogorov Type for Periodic Functions of Two Variables

Ukrainian Mathematical Journal - We obtain a new sharp inequality of the Landau–Kolmogorov type for a periodic function of two variables estimating the convolution of the best uniform...     

A Remark on the Steklov–Poincaré Inequality

Mathematical Notes - In an $$n$$ -dimensional bounded domain $$\Omega_n$$ , $$n\ge 2$$ , we prove the Steklov–Poincaré inequality with the best constant in the case where $$\Omega_n$$ is...     

A Brunn–Minkowski-Type Inequality

We formulate and discuss a conjecture that might strengthen the Brunn–Minkowski inequality.     

A Refinement of the Robertson–Schrödinger Uncertainty Principle and a Hirschman–Shannon Inequality for Wigner Distributions

Journal of Fourier Analysis and Applications - We propose a refinement of the Robertson–Schrödinger uncertainty principle (RSUP) using Wigner distributions. This new principle is...