Sobolev Constant Research Articles

A sharp criterion for zero modes of the Dirac equation

It is shown that \Vert A \Vert_{L^d}^2\ge \frac{d}{d-2}S_{d} is a necessary condition for the existence of a nontrivial solution \psi of the Dirac equation \gamma \cdot (-i\nabla -A)\psi = 0 in d dimensions. Here, S_{d} is the sharp Sobolev constant. If d is odd and \Vert A \Vert_{L^d}^2= \frac{d}{d-2}S_{d} , then there exist vector potentials that allow for zero modes. A complete classification of these vector potentials and their corresponding zero modes is given.

Principal eigenvalues and eigenfunctions for fully nonlinear equations in punctured balls

This paper is devoted to the proof of the existence of the principal eigenvalue and related eigenfunctions for fully nonlinear uniformly elliptic equations posed in a punctured ball, in presence of a singular potential. More precisely, we analyze existence, uniqueness and regularity of solutions (λ¯γ,uγ) of the equationF(D2uγ)+λ¯γuγrγ=0inB(0,1)∖{0},uγ=0on∂B(0,1) where uγ>0 in B(0,1)∖{0} and γ>0. We prove existence of radial solutions which are continuous on B(0,1)‾ in the case γ<2, existence of unbounded solutions in the case γ=2 and a non existence result for γ>2. We also give, in the case of Pucci's operators, the explicit value of λ¯2, which generalizes the Hardy–Sobolev constant for the Laplacian.

Stability with explicit constants of the critical points of the fractional Sobolev inequality and applications to fast diffusion

We study the quantitative stability of critical points of the fractional Sobolev inequality. We show that, for a non-negative function u∈H˙s(RN) whose energy satisfies12SN,sN2s≤‖u‖H˙s(RN)≤32SN,sN2s, where SN,s is the optimal Sobolev constant, the bound‖u−U[z,λ]‖H˙s(RN)≲‖(−Δ)su−u2s⁎−1‖H˙−s(RN), holds for a suitable fractional Talenti bubble U[z,λ]. For functions u which are close to Talenti bubbles, we give the sharp asymptotic value of the implied constant in this inequality. As an application of this, we derive an explicit polynomial extinction rate for positive solutions to a fractional fast diffusion equation.

Global existence, blow-up and mass concentration for the inhomogeneous nonlinear Schrödinger equation with inverse-square potential

&lt;abstract&gt;&lt;p&gt;In the current paper, the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation including inverse-square potential is considered. First, some criteria of global existence and finite-time blow-up in the mass-critical and mass-supercritical settings with $ 0 &amp;lt; c\leq c^{*} $ are obtained. Then, by utilizing the potential well method and the sharp Sobolev constant, the sharp condition of blow-up is derived in the energy-critical case with $ 0 &amp;lt; c &amp;lt; \frac{N^{2}+4N}{(N+2)^{2}}c^{*} $. Finally, we establish the mass concentration property of explosive solutions, as well as the dynamic behaviors of the minimal-mass blow-up solutions in the $ L^{2} $-critical setting for $ 0 &amp;lt; c &amp;lt; c^{*} $, by means of the variational characterization of the ground-state solution to the elliptic equation, scaling techniques and a suitable refined compactness lemma. Our results generalize and supplement the ones of some previous works.&lt;/p&gt;&lt;/abstract&gt;

Existence results for nonlocal problems governed by the regional fractional Laplacian

The aim of the present paper is to study existence results of minimizers of the critical fractional Sobolev constant on bounded domains. Under some values of the fractional parameter we show that the best constant is achieved. If moreover the underlying domain is a ball, we obtain positive radial minimizers for all possible values of the fractional parameter in higher dimension, while we impose a positive mass condition in low dimension.

Coalescing and branching simple symmetric exclusion process

Motivated by kinetically constrained interacting particle systems (KCM), we consider a reversible coalescing and branching simple exclusion process on a general finite graph G=(V,E) dual to the biased voter model on G. Our main goal is tight bounds on its logarithmic Sobolev constant and relaxation time, with particular focus on the delicate slightly supercritical regime in which the equilibrium density of particles tends to zero as |V|→∞. Our results allow us to recover very directly and improve to ℓp-mixing, p≥2, and to more general graphs, the mixing time results of Pillai and Smith for the Fredrickson–Andersen one spin facilitated (FA-1f) KCM on the discrete d-dimensional torus. In view of applications to the more complex FA-jf KCM, j>1, we also extend part of the analysis to an analogous process with a more general product state space.

Ground state for critical elliptic systems with perturbation term of superlinear type

This work deals with the existence of at least one positive ground state solution for a stationary perturbed critical elliptic system with superlinear potential. Our problems involve the critical Sobolev constants which generate the lack of compactness in unbounded domains; we overcome such difficulty by using the concept of the Palais–Smale convergence. We make recourse to the Ekeland variational principle to show that our problem has a positive time-independent solution with positive energy as the total energy of the system.

Trace Hardy-Sobolev-Maz'ya inequalities on half space and sharp constant in dimension two

The main purpose of this paper is to establish trace Hardy-Sobolev-Maz'ya inequalities on half space. In case n=2, we show that the sharp constant coincides with the best trace Sobolev constant. This is an analogous result to that of the sharp constant in the n−12-th order Hardy-Sobolev-Maz'ya inequality in the half space of dimension n when n is odd.

Hypercontractivity and Logarithmic Sobolev Inequality for Non-primitive Quantum Markov Semigroups and Estimation of Decoherence Rates

We generalize the concepts of weak quantum logarithmic Sobolev inequality (LSI) and weak hypercontractivity (HC), introduced in the quantum setting by Olkiewicz and Zegarlinski, to the case of non-primitive quantum Markov semigroups (QMS). The originality of this work resides in that this new notion of hypercontractivity is given in terms of the so-called amalgamatedmathbb {L}_p norms introduced recently by Junge and Parcet in the context of operator spaces theory. We make three main contributions. The first one is a version of Gross’ integration lemma: we prove that (weak) HC implies (weak) LSI. Surprisingly, the converse implication differs from the primitive case as we show that LSI implies HC but with a weak constant equal to the cardinal of the center of the decoherence-free algebra. Building on the first implication, our second contribution is the fact that strong LSI and therefore strong HC do not hold for non-trivially primitive QMS. This implies that the amalgamated mathbb {L}_p norms are not uniformly convex for 1le p le 2. As a third contribution, we derive universal bounds on the (weak) logarithmic Sobolev constants for a QMS on a finite dimensional Hilbert space, using a similar method as Diaconis and Saloff-Coste in the case of classical primitive Markov chains, and Temme, Pastawski and Kastoryano in the case of primitive QMS. This leads to new bounds on the decoherence rates of decohering QMS. Additionally, we apply our results to the study of the tensorization of HC in non-commutative spaces in terms of the completely bounded norms (CB norms) recently introduced by Beigi and King for unital and trace preserving QMS. We generalize their results to the case of a general primitive QMS and provide estimates on the (weak) constants.

Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems

In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015–4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean–Vlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971–1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109–132; Ann. Appl. Probab. 13 (2003) 540–560) (by Bakry–Emery’s technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429–445) (by dissipation of the Wasserstein distance).

Which magnetic fields support a zero mode?

Abstract This paper presents some results concerning the size of magnetic fields that support zero modes for the three-dimensional Dirac equation and related problems for spinor equations. It is a well-known fact that for the Schrödinger equation in three dimensions to have a negative energy bound state, the 3 / 2 {3/2} norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3 / 2 {3/2} norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.

Logarithmic Sobolev inequalities on non-isotropic Heisenberg groups

We study logarithmic Sobolev inequalities with respect to a heat kernel measure on finite-dimensional and infinite-dimensional Heisenberg groups. Such a group is the simplest non-trivial example of a sub-Riemannian manifold. First we consider logarithmic Sobolev inequalities on non-isotropic Heisenberg groups. These inequalities are considered with respect to the hypoelliptic heat kernel measure, and we show that the logarithmic Sobolev constants can be chosen to be independent of the dimension of the underlying space. In this setting, a natural Laplacian is not an elliptic but a hypoelliptic operator. The argument relies on comparing logarithmic Sobolev constants for the three-dimensional non-isotropic and isotropic Heisenberg groups, and tensorization of logarithmic Sobolev inequalities in the sub-Riemannian setting. Furthermore, we apply these results in an infinite-dimensional setting and prove a logarithmic Sobolev inequality on an infinite-dimensional Heisenberg group modeled on an abstract Wiener space.

Convergence of the Ricci flow on asymptotically flat manifolds with integral curvature pinching

We prove a curvature pinching result for the Ricci flow on asymptotically flat manifolds: if an asymptotically flat manifold of dimension $n\geq 3$ has scale-invariant integral norm of curvature sufficiently pinched relative to the inverse of its Sobolev constant, then the Ricci flow starting from this manifold exists for all positive times and converges to flat Euclidean space. In particular our result implies that the initial manifold must have been diffeomorphic to $\mathbb{R}^n$.

Solutions to coupled critical elliptic systems involving attractive Hardy–type terms

In this paper, a system of elliptic equations is studied, which involves coupled critical nonlinearities and attractive Hardy terms. By the local compactness analysis of (PS)c sequence and estimates on extremal functions corresponding to the best Sobolev constant, the existence of Mountain–Pass solutions is proved. Secondly, the asymptotic properties at the origin of solutions to the system are proved by the analytic technics and variational arguments.

Some remarks on the Sobolev inequality in Riemannian manifolds

We investigate Sobolev and Hardy inequalities, specifically weighted Minerbe’s type estimates, in noncompact complete connected Riemannian manifolds whose geometry is described by an isoperimetric profile. In particular, we assume that the manifold satisfies the p p -hyperbolicity property, stated in terms of a necessary integral Dini condition on the isoperimetric profile. Our method seems to us to combine sharply the knowledge of the isoperimetric profile and the optimal Bliss type Hardy inequality depending on the geometry of the manifold. We recover the well known best Sobolev constant in the Euclidean case.

Green's functions of Paneitz and GJMS operators on hyperbolic spaces and sharp Hardy-Sobolev-Maz'ya inequalities on half spaces

Using the Helgason-Fourier analysis techniques on hyperbolic spaces and Green's function estimates, we prove in a unified way that the sharp constant in the n−12-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n coincides with the best n−12-th order Sobolev constant when n is odd and n≥5. We will also establish a lower bound of the coefficient of the Hardy term for the k-th order Hardy-Sobolev-Maz'ya inequality in upper half space in the remaining cases of dimension n and k-th order derivatives. As a consequence, we thus show that the sharp constant in the k-th order Hardy-Sobolev-Maz'ya inequality in the upper half space of dimension n is strictly less than the best k-th order Sobolev constant for all n≥2k+2. Precise expressions and optimal bounds for Green's functions of the operator −ΔH−(n−1)24 on the hyperbolic space Bn and operators of the product form are given, where (n−1)24 is the spectral gap for the Laplacian −ΔH on Bn. Finally, we give the precise expression and optimal pointwise bound of Green's function of the Paneitz and GJMS operators on hyperbolic space in terms of hypergeometric functions. In fact, we will establish the precise formulas of the Green's functions of the operator∏j=0k−1((ν+j)2−(n−1)2/4−ΔH),ν≥0, in terms of hypergeometric function F(a,b;c,z).Our approach to establish the main theorems is using the Helgason-Fourier analysis on hyperbolic spaces and are substantially different from that in dealing with the first order Hardy-Sobolev-Maz'ya inequalities on upper half spaces.

Existence of positive solutions for Brezis-Nirenberg type problems involving an inverse operator

This article concerns the existence of positive solutions for thesecond order equation involving a nonlocal term $$ -\Delta u=\gamma (-\Delta)^{-1} u+|u|^{p-1}u, $$ under Dirichlet boundary conditions. We prove the existence of positive solutions depending on the positive real parameter \(\gamma&gt;0\), and up to the critical value of the exponent \(p\), i.e. when \(1&lt;p\leq 2^*-1\), where \(2^*=\frac{2N}{N-2}\) is the critical Sobolev exponent. For \(p=2^*-1\), this leads us to a Brezis-Nirenberg type problem, cf. \cite{BN}, but, in our particular case, the linear term is a nonlocal term. The effect that this nonlocal term has on the equation changes the dimensions for which the classical technique based on the minimizers of the Sobolev constant, that ensures the existence of positive solution, going from dimensions \(N\geq 4\) in the classical Brezis-Nirenberg problem, to dimensions \(N\geq7\) for this nonlocal problem. For more information see https://ejde.math.txstate.edu/Volumes/2021/52/abstr.html

On the modified logarithmic Sobolev inequality for the heat-bath dynamics for 1D systems

The mixing time of Markovian dissipative evolutions of open quantum many-body systems can be bounded using optimal constants of certain quantum functional inequalities, such as the modified logarithmic Sobolev constant. For classical spin systems, the positivity of such constants follows from a mixing condition for the Gibbs measure, via quasi-factorization results for the entropy. Inspired by the classical case, we present a strategy to derive the positivity of the modified logarithmic Sobolev constant associated to the dynamics of certain quantum systems from some clustering conditions on the Gibbs state of a local, commuting Hamiltonian. In particular we show that for the heat-bath dynamics for 1D systems, the modified logarithmic Sobolev constant is positive under the assumptions of a mixing condition on the Gibbs state and a strong quasi-factorization of the relative entropy.

Hardy–Adams Inequalities on ℍ2 × ℝ n-2

Abstract Let ℍ 2 {\mathbb{H}^{2}} be the hyperbolic space of dimension 2. Denote by M n = ℍ 2 × ℝ n - 2 {M^{n}=\mathbb{H}^{2}\times\mathbb{R}^{n-2}} the product manifold of ℍ 2 {\mathbb{H}^{2}} and ℝ n - 2 ( n ≥ 3 ) {\mathbb{R}^{n-2}(n\geq 3)} . In this paper we establish some sharp Hardy–Adams inequalities on M n {M^{n}} , though M n {M^{n}} is not with strictly negative sectional curvature. We also show that the sharp constant in the Poincaré–Sobolev inequality on M n {M^{n}} coincides with the best Sobolev constant, which is of independent interest.

Energy asymptotics in the three-dimensional Brezis\u2013Nirenberg problem

For a bounded open set Omega subset {mathbb {R}}^3 we consider the minimization problem S(a+ϵV)=inf0≢u∈H01(Ω)∫Ω(|∇u|2+(a+ϵV)|u|2)dx(∫Ωu6dx)1/3\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} S(a+\\epsilon V) = \\inf _{0\\not \\equiv u\\in H^1_0(\\Omega )} \\frac{\\int _\\Omega (|\\nabla u|^2+ (a+\\epsilon V) |u|^2)\\,dx}{(\\int _\\Omega u^6\\,dx)^{1/3}} \\end{aligned}$$\\end{document}involving the critical Sobolev exponent. The function a is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on a and V we compute the asymptotics of S(a+epsilon V)-S as epsilon rightarrow 0+, where S is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to a and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have S(a+epsilon V)<S for all sufficiently small epsilon >0.