Hardy Inequality Research Articles

Extremal problems in geometric function theory

This survey is devoted to a number of achievements in the theory of extremal problems in geometric function theory. The approaches to the solution of problems under consideration and the methods used are based on conformal isomorphisms and on the theory of univalent functions developed since the beginning of the 20th century. Results on integral means of conformal mappings of a disc are presented and, in particular, Dolzenko's inequality for rational functions is extended to arbitrary domains with rectifiable boundaries. Investigations in the field of Bohr-type inequalities are described. An emphasis is made on integral inequalities of Hardy and Rellich type, in which the analytic properties of inequalities are intertwined with geometric characteristics of the boundaries of domains. Results related to the solution of the Vuorinen problem on the behaviour of conformal moduli under unlimited dilations of the plane are presented. Formulae for the variation of Robin capacity are obtained. One-parameter families of rational and elliptic functions whose critical values vary in accordance with a prescribed law are characterized. The last results on Smale's conjecture and Smale's dual conjecture are described. Bibliography: 149 titles.

The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $ H^{s}(\mathbb R^{n}) $

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation$ iu_{t} +\Delta u = |x|^{-b} f(u), \; u(0) = u_{0} \in H^{s} (\mathbb R^{n} ), $where $ n\ge3 $, $ 1< s<\frac{n}{2} $, $ 0<b<2 $ and $ f(u) $ is a nonlinear function that behaves like $ \lambda |u|^{\sigma } u $ with $ \lambda \in \mathbb C $ and $ \sigma = \frac{4-2b}{n-2s} $. We establish the local well-posedness as well as the small data global well-posedness and scattering in $ H^{s} (\mathbb R^{n} ) $ with $ 1<s<\frac{n}{2} $ for the critical INLS equation under some assumption on $ b $. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.

On the constant in the Hardy inequality for finite sequences

On the constant in the Hardy inequality for finite sequences

Sharp weak bounds for discrete Hardy operator on discrete central Morrey spaces

<abstract><p>In this note, we introduce the discrete (weak) central Morrey spaces, which are central versions of discrete (weak) Morrey spaces. The sharp bounds for discrete Hardy operator from discrete central Morrey spaces to discrete weak central Morrey spaces are proven to be equal to 1. As an application, we obtain the weak version of the well-known discrete Hardy inequality.</p></abstract>

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Hardy inequalities on metric measure spaces, III: the case q ≤ p ≤ 0 and applications

In this paper, we obtain a reverse version of the integral Hardy inequality on metric measure space with two negative exponents. For applications we show the reverse Hardy–Littlewood–Sobolev and the Stein–Weiss inequalities with two negative exponents on homogeneous Lie groups and with arbitrary quasi-norm, the result of which appears to be new in the Euclidean space. This work further complements the ranges of p and q (namely, q ≤ p < 0 ) considered in the work of Ruzhansky & Verma (Ruzhansky & Verma 2019 Proc. R. Soc. A 475 , 20180310 ( doi:10.1098/rspa.2018.0310 ); Ruzhansky & Verma. 2021 Proc. R. Soc. A 477 , 20210136 ( doi:10.1098/rspa.2021.0136 )), which treated the cases 1 < p ≤ q < ∞ and p > q , respectively.

Multipolar potentials and weighted Hardy inequalities

In this paper we state the following weighted Hardy type inequality for any functions $ \varphi $ in a weighted Sobolev space and for weight functions $ \mu $ of a quite general type \begin{document}$ \begin{align*} c_{N, \mu} \int_{ \mathbb{R}^N}V\, \varphi^2\mu(x)dx\le \int_{ \mathbb{R}^N}|\nabla \varphi|^2\mu(x)dx +C_\mu \int_{ \mathbb{R}^N}W \varphi^2\mu(x)dx, \end{align*} $\end{document} where $ V $ is a multipolar potential and $ W $ is a bounded function from above depending on $ \mu $. Our method is based on introducing a suitable vector-valued function and an integral identity that we state in the paper. We prove that the constant $ c_{N, \mu} $ in the estimate is optimal by building a suitable sequence of functions.

Weighted Hardy inequality with two-dimensional rectangular operator: the case q<p

Weighted Hardy inequality with two-dimensional rectangular operator: the case q<p

Integral Hardy inequalities, their generalizations and related inequalities

Integral Hardy inequalities, their generalizations and related inequalities

The infimum eigenvalue for degenerate p(x)-biharmonic operator with the Hardy potentiel

The aim of this article is to study the existence of at least one unbounded nondecreasing sequence of nonnegative eigenvalues (λk)k≥1 for a class of elliptic Navier boundary value problems involving the degenerate p(·)-biharmonic operator with q(x)-Hardy inequality by using the variational technique based on the Ljusternik-Schnirelmann theory on C1-manifolds and the theory of the variable exponent Lebesgue spaces. Also, we obtain the positivity of the infimum eigenvalue for the problem.

Dynamic Inequalities of Two-Dimensional Hardy Type via Alpha-Conformable Derivatives on Time Scales

We established some new α-conformable dynamic inequalities of Hardy–Knopp type. Some new generalizations of dynamic inequalities of α-conformable Hardy type in two variables on time scales are established. Furthermore, we investigated Hardy’s inequality for several functions of α-conformable calculus. Our results are proved by using two-dimensional dynamic Jensen’s inequality and Fubini’s theorem on time scales. When α=1, then we obtain some well-known time-scale inequalities due to Hardy. As special cases, we derived Hardy’s inequality for T=R,T=Z and T=hZ. Symmetry plays an essential role in determining the correct methods to solve dynamic inequalities.

Probability theoretic generalizations of Hardy’s and Copson’s inequality

A short proof of the classic Hardy inequality is presented for p-norms with p>1. Along the lines of this proof a sharpened version is proved of a recent generalization of Hardy’s inequality in the terminology of probability theory. A probability theoretic version of Copson’s inequality is discussed as well. Also for 0<p<1 probability theoretic generalizations of the Hardy and the Copson inequality are proved.

Some Inequalities of Hardy Type Related to Witten–Laplace Operator on Smooth Metric Measure Spaces

A complete Riemannian manifold equipped with some potential function and an invariant conformal measure is referred to as a complete smooth metric measure space. This paper generalizes some integral inequalities of the Hardy type to the setting of a complete non-compact smooth metric measure space without any geometric constraint on the potential function. The adopted approach highlights some criteria for a smooth metric measure space to admit Hardy inequalities related to Witten and Witten p-Laplace operators. The results in this paper complement in several aspect to those obtained recently in the non-compact setting.

On the compactness of the non-radial Sobolev space

In this paper, we give the affirmative answer of the question in [17], which is a compactness result of the non-radial Sobolev space. As an application, we show the existence of an extremal function of the critical Hardy inequality under spherical average zero. In Appendix, we give an alternative proof of Hardy type inequalities under spherical average zero.

An Improved One-Dimensional Hardy Inequality

An Improved One-Dimensional Hardy Inequality

Hardy’s Identities and Inequalities on Cartan-Hadamard Manifolds

We study the Hardy identities and inequalities on Cartan-Hadamard manifolds using the notion of a Bessel pair. These Hardy identities offer significantly more information on the existence/nonexistence of the extremal functions of the Hardy inequalities. These Hardy inequalities are in the spirit of Brezis-Vázquez in the Euclidean spaces. As direct consequences, we establish several Hardy type inequalities that provide substantial improvements as well as simple understandings to many known Hardy inequalities and Hardy-Poincaré-Sobolev type inequalities on hyperbolic spaces in the literature.

Functional inequalities and applications to doubly nonlinear diffusion equations

Abstract We study weighted inequalities of Hardy and Hardy–Poincaré type and find necessary and sufficient conditions on the weights so that the considered inequalities hold. Examples with the optimal constants are shown. Such inequalities are then used to quantify the convergence rate of solutions to doubly nonlinear fast diffusion equation towards the Barenblatt profile.

The Saint‐Venant type isoperimetric inequalities for assessing saturated water storage in lacunary shallow perched aquifers

AbstractThe Dupuit‐Forchheimer (DF) approximation for unconfined groundwater flows is reduced to 2D Poisson's equation (PE), the right hand side of which involves an intensive evapotranspiration (ET) rate. A shallow water table dips inward from a constant piezometric head boundary (closed curve) such that a “dry gap,” demarcated by another closed curve (unknown front), may emerge. Apriori estimates of the volume of the saturated zone and of the “dry gap” area are important for water resources management in drylands. We use the results from the theory of linear elasticity, torsion of elastic bars, for which PE is solved for the Prandtl function. Using the Poincaré metrics with the Gaussian constant curvature , isoperimetric inequalities are obtained for steady‐state DF flows. Conformal moments and confromal radii of the domains and 2‐D Hardy's type inequalities in domains, modeling the Saint Venant bar torsion problem, are involved in the obtained estimates. The studied boundary value problems (BVPs) are nonlinear for ET rates depending on the depth of the water table. In the DF model, the vadose zone (VZ) is considered as a “distributed sink” (similar to a standard “distributed source,” which models recharge to the water table in humid climates). The analytical VZ is collated with one obtained from a numerical solution to BVP for Richards’ equation in a 3‐D saturated‐unsaturated flow. BVPs for Richards’ equation in cylindrical domains, solved by HYDRUS software, give the pressure head, moisture content, Darcian velocity fields, and streamlines.

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

AbstractIn this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space . The space is the unit disc of endowed with the Riemannian metric g given by , where and , if and , if . Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis–Marcus, which allows us to overcome the difficulty involving the singularities.

Hardy inequality for domains with a geometric boundary condition and applications

In this paper, we state a Hardy inequality for domains with a geometric boundary condition. As a consequence, we prove a weighted Trudinger–Moser inequality. After that, we apply our results to investigate the existence of solutions for a class of quasilinear elliptic equations with Neumann boundary condition and non-linearities with critical exponential growth.