The Rellich Inequality

2018 ◽  
pp. 235-247
Author(s):  
David E. Edmunds ◽  
W. Desmond Evans
Keyword(s):  
Rellich Inequality

The Rellich Inequality

2015 ◽  
pp. 213-249
Author(s):  
Alexander A. Balinsky ◽  
W. Desmond Evans ◽  
Roger T. Lewis
Keyword(s):  
Rellich Inequality

scholarly journals Remarks on the Rellich inequality

2016 ◽  
Vol 286 (3-4) ◽  
pp. 1367-1373 ◽  
Author(s):  
Shuji Machihara ◽  
Tohru Ozawa ◽  
Hidemitsu Wadade
Keyword(s):  
Rellich Inequality

scholarly journals Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities

2020 ◽  
pp. 2050024 ◽  
Author(s):  
Andrei Velicu
Keyword(s):  
Hardy Inequality ◽  
Root Systems ◽  
Dunkl Operators ◽  
Hardy Inequalities ◽  
One Dimensional ◽  
Hardy Type Inequalities ◽  
Rellich Inequality ◽  
Nirenberg Inequality ◽  
Special Case ◽  
The Hardy Inequality

In this paper, we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, [Formula: see text] 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 [Formula: see text] improve some well-known results.

The Rellich inequality

2016 ◽  
Vol 29 (3) ◽  
pp. 511-530 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Rellich Inequality

scholarly journals A multilinear Rellich inequality

2021 ◽  
pp. 265-274
Author(s):  
David E. Edmunds ◽  
Alexander Meskhi
Keyword(s):  
Rellich Inequality

New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements

2019 ◽  
Vol 150 (6) ◽  
pp. 2952-2981 ◽  
Author(s):  
Van Hoang Nguyen
Keyword(s):  
Sectional Curvature ◽  
Hyperbolic Space ◽  
Hardy Inequality ◽  
Hadamard Manifolds ◽  
Hardy Inequalities ◽  
Hardy Type Inequalities ◽  
Rellich Inequality ◽  
Hardy Type

In this paper, we prove several new Hardy type inequalities (such as the weighted Hardy inequality, weighted Rellich inequality, critical Hardy inequality and critical Rellich inequality) related to the radial derivation (i.e., the derivation along the geodesic curves) on the Cartan–Hadamard manifolds. By Gauss lemma, our new Hardy inequalities are stronger than the classical ones. We also establish the improvements of these inequalities in terms of sectional curvature of the underlying manifolds which illustrate the effect of curvature to these inequalities. Furthermore, we obtain some improvements of Hardy and Rellich inequalities on the hyperbolic space ℍn. Especially, we show that our new Rellich inequalities are indeed stronger than the classical ones on the hyperbolic space ℍn.

scholarly journals The Hardy–Rellich inequality for polyharmonic operators

1999 ◽  
Vol 129 (4) ◽  
pp. 825-839 ◽  
Author(s):  
Mark P. Owen
Keyword(s):  
Boundary Conditions ◽  
Hardy Inequality ◽  
Dirichlet Boundary ◽  
Higher Order ◽  
Dirichlet Boundary Conditions ◽  
Spectral Information ◽  
Dirichlet Laplacian ◽  
Rellich Inequality

The Hardy-Rellich inequality given here generalizes a Hardy inequality of Davies, from the case of the Dirichlet Laplacian of a region Ω ⊆ ℝN to that of the higher-order polyharmonic operators with Dirichlet boundary conditions. The inequality yields some immediate spectral information for the polyharmonic operators and also bounds on the trace of the associated semigroups and resolvents.

Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential

2019 ◽  
Vol 9 (1) ◽  
pp. 1130-1144 ◽  
Author(s):  
Abdelouahed El Khalil ◽  
Mohamed Laghzal ◽  
My Driss Morchid Alaoui ◽  
Abdelfattah Touzani
Keyword(s):  
Eigenvalue Problem ◽  
Real Number ◽  
Distance Function ◽  
Positive Real Number ◽  
Nonlinear Eigenvalue Problem ◽  
Biharmonic Operator ◽  
Hardy Potential ◽  
Positive Real ◽  
Singular Problems ◽  
Rellich Inequality

Abstract In this paper, we consider the nonlinear eigenvalue problem: $$\begin{array}{} \displaystyle \begin{cases} {\it\Delta}(|{\it\Delta} u|^{p(x)-2}{\it\Delta} u)= \lambda \frac{|u|^{q(x)-2}u}{{\delta(x)}^{2q(x)}} \;\; \mbox{in}\;\; {\it\Omega}, \\ u\in W_0^{2,p(x)}({\it\Omega}), \end{cases} \end{array}$$ where Ω is a regular bounded domain of ℝN, δ(x) = dist(x, ∂Ω) the distance function from the boundary ∂Ω, λ is a positive real number, and functions p(⋅), q(⋅) are supposed to be continuous on Ω satisfying $$\begin{array}{} \displaystyle 1 \lt \min_{\overline{{\it\Omega} }}\,q\leq \max_{\overline{{\it\Omega}}}\,q \lt \min_{\overline{{\it\Omega} }}\,p \leq \max_{\overline{{\it\Omega}}}\,p \lt \frac{N}{2} \mbox{ and } \max_{\overline{{\it\Omega}}}\,q \lt p_2^*:= \frac{Np(x)}{N-2p(x)} \end{array}$$ for any x ∈ Ω. We prove the existence of at least one non-decreasing sequence of positive eigenvalues. Moreover, we prove that sup Λ = +∞, where Λ is the spectrum of the problem. Furthermore, we give a proof of positivity of inf Λ > 0 provided that Hardy-Rellich inequality holds.

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