Narrow Results

In this paper we derive both local and global geometric inequalities on general Riemannnian and Finsler manifolds and prove generalized Caffarelli–Kohn–Nirenberg type and Hardy type inequalities on Finsler manifolds, illuminating curvatures of both Riemannian and Finsler manifolds influence geometric inequalities.

In this paper, we establish some weighted Hardy and Rellich inequalities and discuss its best constants on the Heisenberg group. Moreover, we also present a class of higher-order weighted Hardy–Rellich inequalities with the remainder term.

We present and analyze two mathematical models for the self-consistent quantum transport of electrons in a graphene layer. We treat two situations. First, when the particles can move in all the plane ℝ², the model takes the form of a system of massless Dirac equations coupled together by a self-consistent potential, which is the trace in the plane of the graphene of the 3D Poisson potential associated to surface densities. In this case, we prove local in time existence and uniqueness of a solution in H^s(ℝ²), for which includes in particular the energy space H^1/2(ℝ²). The main tools that enable to reach are the dispersive Strichartz estimates that we generalized here for mixed quantum states. Second, we consider a situation where the particles are constrained in a regular bounded domain Ω. In order to take into account Dirichlet boundary conditions which are not compatible with the Dirac Hamiltonian H₀, we propose a different model built on a modified Hamiltonian displaying the same energy band diagram as H₀ near the Dirac points. The well-posedness of the system in this case is proved in , the domain of the fractional order Dirichlet Laplacian operator, for .

We establish the analog Bliss and Hardy inequalities with sharp constant involving exponential weight function. One special case of the inequalities (for n = 2) leads to a direct proof of Onofri inequality on S².

Let Ω be a smooth bounded domain in ℝ^N with N ≥ 1. In this paper we study the Hardy–Poincaré inequality with weight function singular at the boundary of Ω. In particular we provide sufficient and necessary conditions on the existence of minimizers.

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain the sharp constants of Hardy and Rellich inequalities related to the geodesic distance on M. Furthermore, if M is with strictly negative curvature, we show that the L^p Hardy inequalities can be globally refined by adding remainder terms like the Brezis–Vázquez improvement in case p ≥ 2, which is contrary to the case of Euclidean spaces.

Let Ω ⊂ ℝ^N be a bounded regular domain of ℝ^N and 1 < p < ∞. The paper is divided into two main parts. In the first part, we prove the following improved Hardy inequality for convex domains. Namely, for all , we have

In this paper, we study the optimization problem

For a bounded domain $\mathrm{\Omega }$ containing all the singularities in the interior, we prove that ${\mu }^{\star }(\mathrm{\Omega })>{\mu }^{\star }({ℝ}^N)$ when $n\ge 3$ and ${\mu }^{\star }(\mathrm{\Omega })={\mu }^{\star }({ℝ}^N)$ when $n=2$ (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 ${\mu }^{\star }({ℝ}^N)={(N-2)}^2/n^2)$.

In the situation when all the poles are located on the boundary, we show that ${\mu }^{\star }(\mathrm{\Omega })=N^2/n^2$ if $\mathrm{\Omega }$ 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 ${\mu }^{\star }(\mathrm{\Omega })$ is attained in $H_0^1(\mathrm{\Omega })$ when $\mathrm{\Omega }$ is a ball and $n\ge 3$. Besides, ${\mu }^{\star }(\mathrm{\Omega })$ is attained in ${{𝒟}}^{1,2}(\mathrm{\Omega })$ when $\mathrm{\Omega }$ is the exterior of a ball with $N\ge 3$ and $n\ge 3$ whereas in the case of a half-space ${\mu }^{\star }(\mathrm{\Omega })$ is attained in ${{𝒟}}^{1,2}(\mathrm{\Omega })$ when $n\ge 3$.

We also analyze the critical constants in the so-called weak Hardy inequality which characterizes the range of ${\mu }^{\prime }s$ ensuring the existence of a lower bound for the spectrum of the Schrödinger operator $-\mathrm{\Delta }-\mu V$. In the context of both interior and boundary singularities, we show that the critical constants in the weak Hardy inequality are ${(N-2)}^2/(4n-4)$ and $N^2/(4n-4)$, respectively.

We establish sharp remainder terms of the $L^2$-Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups, yielding the inequalities with best constants. Our methods also give new sharp Caffarelli–Kohn–Nirenberg-type inequalities in ${ℝ}^n$ with arbitrary quasi-norms. We also present explicit examples to illustrate our results for different weights and in abelian cases.

In this paper, we derive the following Leray–Trudinger type inequality on a bounded domain $\mathrm{\Omega }$ in ${ℝ}^n$ containing the origin.

We study the behavior of Hardy-weights for a class of variational quasilinear elliptic operators of $p$-Laplacian type. In particular, we obtain necessary sharp decay conditions at infinity on the Hardy-weights in terms of their integrability with respect to certain integral weights. Some of the results are extended also to nonsymmetric linear elliptic operators. Applications to various examples are discussed as well.

In this paper, we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, $L^p$ inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli–Kohn–Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalized root systems are proved, which in the particular case of root systems $A_{N-1}$ improve some well-known results.

We show a symmetric Markov diffusion semigroup satisfies a weighted contractivity condition if and only if a $L^2$-Hardy inequality holds, and we give a Bakry–Émery-type criterion for the former. We then give some applications.

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an $L^p$-function inside the domain of definition but as smooth as a $W^{s,p}$-function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces $W^{s,p}(\mathrm{\Omega })$. Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space $W^{s,p}(\mathrm{\Omega })$. The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition.

We construct the Sobolev type space with the finite norm

In this note we consider the Dirichlet Laplacian in twisted three-dimensional tubes. In particular we study a related Hardy type inequality, which has been establish in previous works, and show how such an inequality can be improved. We also discuss the optimality of the corresponding integral weight.