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Journal articles on the topic 'Rellich inequality'

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Published: 4 June 2021
Last updated: 28 January 2023

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1

Edmunds, D. E., and W. D. Evans. "The Rellich inequality." Revista Matemática Complutense 29, no. 3 (May 13, 2016): 511–30. http://dx.doi.org/10.1007/s13163-016-0200-7.

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2

Nguyen, Van Hoang. "New sharp Hardy and Rellich type inequalities on Cartan–Hadamard manifolds and their improvements." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 6 (August 23, 2019): 2952–81. http://dx.doi.org/10.1017/prm.2019.37.

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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.
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3

Edmunds, David E., and Alexander Meskhi. "A multilinear Rellich inequality." Mathematical Inequalities & Applications, no. 1 (2021): 265–74. http://dx.doi.org/10.7153/mia-2021-24-19.

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4

Cassano, Biagio, Lucrezia Cossetti, and Luca Fanelli. "Improved Hardy-Rellich inequalities." Communications on Pure & Applied Analysis 21, no. 3 (2022): 867. http://dx.doi.org/10.3934/cpaa.2022002.

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<p style='text-indent:20px;'>We investigate Hardy-Rellich inequalities for perturbed Laplacians. In particular, we show that a non-trivial angular perturbation of the free operator typically improves the inequality, and may also provide an estimate which does not hold in the free case. The main examples are related to the introduction of a magnetic field: this is a manifestation of the diamagnetic phenomenon, which has been observed by Laptev and Weidl in [<xref ref-type="bibr" rid="b21">21</xref>] for the Hardy inequality, later by Evans and Lewis in [<xref ref-type="bibr" rid="b9">9</xref>] for the Rellich inequality; however, to the best of our knowledge, the so called Hardy-Rellich inequality has not yet been investigated in this regards. After showing the optimal inequality, we prove that the best constant is not attained by any function in the domain of the estimate.</p>
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5

Machihara, Shuji, Tohru Ozawa, and Hidemitsu Wadade. "Remarks on the Rellich inequality." Mathematische Zeitschrift 286, no. 3-4 (November 7, 2016): 1367–73. http://dx.doi.org/10.1007/s00209-016-1805-8.

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6

Yin, Songting. "A Sharp Rellich Inequality on the Sphere." Mathematics 6, no. 12 (November 27, 2018): 288. http://dx.doi.org/10.3390/math6120288.

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7

Owen, Mark P. "The Hardy–Rellich inequality for polyharmonic operators." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 129, no. 4 (1999): 825–39. http://dx.doi.org/10.1017/s0308210500013160.

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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.
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8

Evans, W. D., and R. T. Lewis. "On the Rellich inequality with magnetic potentials." Mathematische Zeitschrift 251, no. 2 (May 31, 2005): 267–84. http://dx.doi.org/10.1007/s00209-005-0798-5.

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9

Anoop, T. V., Ujjal Das, and Abhishek Sarkar. "On the generalized Hardy-Rellich inequalities." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 2 (January 26, 2019): 897–919. http://dx.doi.org/10.1017/prm.2018.128.

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AbstractIn this paper, we look for the weight functions (say g) that admit the following generalized Hardy-Rellich type inequality: $$\int_\Omega g (x)u^2 dx \les C\int_\Omega \vert \Delta u \vert ^2 dx,\quad \forall u\in {\rm {\cal D}}_0^{2,2} (\Omega ),$$for some constant C > 0, where Ω is an open set in ℝN with N ⩾ 1. We find various classes of such weight functions, depending on the dimension N and the geometry of Ω. Firstly, we use the Muckenhoupt condition for the one-dimensional weighted Hardy inequalities and a symmetrization inequality to obtain admissible weights in certain Lorentz-Zygmund spaces. Secondly, using the fundamental theorem of integration we obtain the weight functions in certain weighted Lebesgue spaces. As a consequence of our results, we obtain simple proofs for the embeddings of ${\cal D}_0^{2,2} $ into certain Lorentz-Zygmund spaces proved by Hansson and later by Brezis and Wainger.
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10

Cazacu, Cristian. "A new proof of the Hardy–Rellich inequality in any dimension." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 6 (August 19, 2019): 2894–904. http://dx.doi.org/10.1017/prm.2019.50.

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The Hardy-Rellich inequality in the whole space with the best constant was firstly proved by Tertikas and Zographopoulos in Adv. Math. (2007) in higher dimensions N ⩾ 5. Then it was extended to lower dimensions N ∈ {3, 4} by Beckner in Forum Math. (2008) and Ghoussoub-Moradifam in Math. Ann. (2011) by applying totally different techniques.In this note, we refine the method implemented by Tertikas and Zographopoulos, based on spherical harmonics decomposition, to give an easy and compact proof of the optimal Hardy–Rellich inequality in any dimension N ⩾ 3. In addition, we provide minimizing sequences which were not explicitly mentioned in the quoted papers in lower dimensions N ∈ {3, 4}, emphasizing their symmetry breaking. We also show that the best constant is not attained in the proper functional space.
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11

Xiao, Ying-Xiong, and Qiao-Hua Yang. "An Improved Hardy-Rellich Inequality with Optimal Constant." Journal of Inequalities and Applications 2009, no. 1 (2009): 610530. http://dx.doi.org/10.1155/2009/610530.

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12

Jin, Yongyang, and Shoufeng Shen. "Weighted Hardy and Rellich inequality on Carnot groups." Archiv der Mathematik 96, no. 3 (February 26, 2011): 263–71. http://dx.doi.org/10.1007/s00013-011-0220-y.

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13

Duy, N. T., and L. L. Phi. "A Note on the Rellich Inequality on Half-Space." Analysis Mathematica 47, no. 3 (July 23, 2021): 559–67. http://dx.doi.org/10.1007/s10476-021-0095-5.

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14

Nasibullin, Ramil. "A geometrical version of Hardy-Rellich type inequalities." Mathematica Slovaca 69, no. 4 (August 27, 2019): 785–800. http://dx.doi.org/10.1515/ms-2017-0268.

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Abstract We obtained a version of Hardy-Rellich type inequality in a domain Ω ∈ ℝn which involves the distance to the boundary, the diameter and the volume of Ω. Weight functions in the inequalities depend on the “mean-distance” function and on the distance function to the boundary of Ω. The proved inequalities connect function to first and second order derivatives.
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15

Dai, Feng, and Yuan Xu. "The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere." Constructive Approximation 40, no. 1 (April 18, 2014): 141–71. http://dx.doi.org/10.1007/s00365-014-9235-5.

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16

Jin, Yongyang, and Yazhou Han. "Weighted Rellich Inequality on H-Type Groups and Nonisotropic Heisenberg Groups." Journal of Inequalities and Applications 2010, no. 1 (2010): 158281. http://dx.doi.org/10.1155/2010/158281.

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17

Duy, Nguyen Tuan, Nguyen Lam, and Le Long Phi. "A note on the second order geometric Rellich inequality on half-space." Monatshefte für Mathematik 195, no. 2 (March 25, 2021): 233–48. http://dx.doi.org/10.1007/s00605-020-01490-9.

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18

Dai, Feng, and Yuan Xu. "Erratum to: The Hardy–Rellich Inequality and Uncertainty Principle on the Sphere." Constructive Approximation 42, no. 1 (March 12, 2015): 181–82. http://dx.doi.org/10.1007/s00365-015-9278-2.

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19

Berchio, Elvise, Debdip Ganguly, Gabriele Grillo, and Yehuda Pinchover. "An optimal improvement for the Hardy inequality on the hyperbolic space and related manifolds." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 150, no. 4 (February 1, 2019): 1699–736. http://dx.doi.org/10.1017/prm.2018.139.

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AbstractWe prove optimal improvements of the Hardy inequality on the hyperbolic space. Here, optimal means that the resulting operator is critical in the sense of Devyver, Fraas, and Pinchover (2014), namely the associated inequality cannot be further improved. Such inequalities arise from more general, optimal ones valid for the operator $ P_{\lambda }:= -\Delta _{{\open H}^{N}} - \lambda $ where 0 ⩽ λ ⩽ λ1(ℍN) and λ1(ℍN) is the bottom of the L2 spectrum of $-\Delta _{{\open H}^{N}} $, a problem that had been studied in Berchio, Ganguly, and Grillo (2017) only for the operator $P_{\lambda _{1}({\open H}^{N})}$. A different, critical and new inequality on ℍN, locally of Hardy type is also shown. Such results have in fact greater generality since they are proved on general Cartan-Hadamard manifolds under curvature assumptions, possibly depending on the point. Existence/nonexistence of extremals for the related Hardy-Poincaré inequalities are also proved using concentration-compactness technique and a Liouville comparison theorem. As applications of our inequalities, we obtain an improved Rellich inequality and we derive a quantitative version of Heisenberg-Pauli-Weyl uncertainty principle for the operator $P_\lambda.$
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20

Cassano, Biagio, Matteo Gallone, and Fabio Pizzichillo. "Dirac–Coulomb operators with infinite mass boundary conditions in sectors." Journal of Mathematical Physics 63, no. 7 (July 1, 2022): 071503. http://dx.doi.org/10.1063/5.0089526.

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We investigate the properties of self-adjointness of a two-dimensional Dirac operator on an infinite sector with infinite mass boundary conditions and in the presence of a Coulomb-type potential with the singularity placed on the vertex. In the general case, we prove the appropriate Dirac–Hardy inequality and exploit the Kato–Rellich theory. In the explicit case of a Coulomb potential, we describe the self-adjoint extensions for all the intensities of the potential relying on a radial decomposition in partial wave subspaces adapted to the infinite-mass boundary conditions. Finally, we integrate our results, giving a description of the spectrum of these operators.
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21

ADIMURTHI. "BEST CONSTANTS AND POHOZAEV IDENTITY FOR HARDY–SOBOLEV-TYPE OPERATORS." Communications in Contemporary Mathematics 15, no. 03 (May 19, 2013): 1250050. http://dx.doi.org/10.1142/s0219199712500502.

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This paper is threefold. Firstly, we reformulate the definition of the norm induced by the Hardy inequality (see [J. L. Vázquez and N. B. Zographopoulos, Functional aspects of the Hardy inequality. Appearance of a hidden energy, preprint (2011); http://arxiv. org/abs/1102.5661]) to more general elliptic and sub-elliptic Hardy–Sobolev-type operators. Secondly, we derive optimal inequalities (see [C. Cowan, Optimal inequalities for general elliptic operator with improvement, Commun. Pure Appl. Anal.9(1) (2010) 109–140; N. Ghoussoub and A. Moradifam, Bessel pairs and optimal Hardy and Hardy–Rellich inequalities, Math. Ann.349(1) (2010) 1–57 (electronic)]) for multiparticle systems in ℝN and Heisenberg group. In particular, we provide a direct proof of an optimal inequality with multipolar singularities shown in [R. Bossi, J. Dolbeault and M. J. Esteban, Estimates for the optimal constants in multipolar Hardy inequalities for Schrödinger and Dirac operators, Commun. Pure Appl. Anal.7(3) (2008) 533–562]. Finally, we prove an approximation lemma which allows to show that the domain of the Dirichlet–Laplace operator is dense in the domain of the corresponding Hardy operators. As a consequence, in some particular cases, we justify the Pohozaev-type identity for such operators.
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22

Khalil, Abdelouahed El, Mohamed Laghzal, My Driss Morchid Alaoui, and Abdelfattah Touzani. "Eigenvalues for a class of singular problems involving p(x)-Biharmonic operator and q(x)-Hardy potential." Advances in Nonlinear Analysis 9, no. 1 (November 7, 2019): 1130–44. http://dx.doi.org/10.1515/anona-2020-0042.

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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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23

Bennett, Donna Marie. "An Extension of Rellich's Inequality." Proceedings of the American Mathematical Society 106, no. 4 (August 1989): 987. http://dx.doi.org/10.2307/2047283.

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24

Bennett, Donna Marie. "An extension of Rellich’s inequality." Proceedings of the American Mathematical Society 106, no. 4 (April 1, 1989): 987. http://dx.doi.org/10.1090/s0002-9939-1989-0955455-6.

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25

PACHPATTE, B. G. "ON SOME EXTENSIONS OF RELLICH'S INEQUALITY." Tamkang Journal of Mathematics 22, no. 3 (September 1, 1991): 259–65. http://dx.doi.org/10.5556/j.tkjm.22.1991.4609.

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26

Fontes, Fernando Galaz. "On Friedrichs inequality and Rellich's theorem." Journal of Mathematical Analysis and Applications 145, no. 2 (January 1990): 516–23. http://dx.doi.org/10.1016/0022-247x(90)90417-e.

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27

Adachi, T., K. Itakura, K. Ito, and E. Skibsted. "New methods in spectral theory of N-body Schrödinger operators." Reviews in Mathematical Physics 33, no. 05 (January 30, 2021): 2150015. http://dx.doi.org/10.1142/s0129055x2150015x.

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We develop a new scheme of proofs for spectral theory of the [Formula: see text]-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem which is unified with exponential decay estimates studied previously only for [Formula: see text]-eigenfunctions. Each pair-potential is a sum of a long-range term with first-order derivatives, a short-range term without derivatives and a singular term of operator- or form-bounded type, and the setup includes hard-core interaction. Our proofs consist of a systematic use of commutators with ‘zeroth order’ operators. In particular, they do not rely on Mourre’s differential inequality technique.
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28

Tung, Chia-chi, and Pier Domenico Lamberti. "On Rellich's Lemma, the Poincaré inequality, and Friedrichs extension of an operator on complex spaces." Cubo (Temuco) 23, no. 2 (August 2021): 265–85. http://dx.doi.org/10.4067/s0719-06462021000200265.

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29

Gupta, Shubham. "Hardy and Rellich inequality on lattices." Calculus of Variations and Partial Differential Equations 62, no. 3 (January 11, 2023). http://dx.doi.org/10.1007/s00526-022-02407-0.

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AbstractIn this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice $$\mathbb {Z}^d$$ Z d as $$d \rightarrow \infty $$ d → ∞ . In the process, we proved some Hardy-type inequalities for the operators $$\Delta ^m$$ Δ m and $$\nabla (\Delta ^m)$$ ∇ ( Δ m ) for non-negative integers m on a d dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as d and $$d^2$$ d 2 respectively as $$ d \rightarrow \infty $$ d → ∞ .
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30

Edmunds, David, and Alexander Meskhi. "On the Rellich inequality in L p(·)(a,b)." Georgian Mathematical Journal, April 6, 2018. http://dx.doi.org/10.1515/gmj-2018-0024.

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31

Velicu, Andrei. "Hardy-type inequalities for Dunkl operators with applications to many-particle Hardy inequalities." Communications in Contemporary Mathematics, June 15, 2020, 2050024. http://dx.doi.org/10.1142/s0219199720500248.

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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.
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32

Hamamoto, Naoki. "Sharp Rellich-Leray inequality for axisymmetric divergence-free vector fields." Calculus of Variations and Partial Differential Equations 58, no. 4 (July 26, 2019). http://dx.doi.org/10.1007/s00526-019-1592-2.

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33

Hamamoto, Naoki. "Sharp Rellich–Leray inequality with any radial power weight for solenoidal fields." Calculus of Variations and Partial Differential Equations 60, no. 2 (April 2021). http://dx.doi.org/10.1007/s00526-021-01956-0.

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34

"Weighted Sobolev spaces and exterior problems for the Helmholtz equation." Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 410, no. 1839 (April 8, 1987): 373–83. http://dx.doi.org/10.1098/rspa.1987.0044.

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Weighted Sobolev spaces are used to settle questions of existence and uniqueness of solutions to exterior problems for the Helmholtz equation. Furthermore, it is shown that this approach can cater for inhomogeneous terms in the problem that are only required to vanish asymptotically at infinity. In contrast to the Rellich–Sommerfeld radiation condition which, in a Hilbert space setting, requires that all radiating solutions of the Helmholtz equation should satisfy a condition of the form ( ∂ / ∂ r − i k ) u ∈ L 2 ( Ω ) , r = | x | ∈ Ω ⊂ R n , it is shown here that radiating solutions satisfy a condition of the form ( 1 + r ) − 1 2 ( ln ( e + r ) ) − 1 2 δ u ∈ L 2 ( Ω ) , 0 < δ < 1 2 , and, moreover, such solutions satisfy the classical Sommerfeld condition u = O ( r − 1 2 ( n − 1 ) ) , r → ∞ . Furthermore, the approach avoids many of the difficulties usually associated with applications of the Poincaré inequality and the Sobolev embedding theorems.
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35

Brown, Donald L., and Dietmar Gallistl. "Multiscale Sub-grid Correction Method for Time-Harmonic High-Frequency Elastodynamics with Wave Number Explicit Bounds." Computational Methods in Applied Mathematics, March 12, 2022. http://dx.doi.org/10.1515/cmam-2022-0041.

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Abstract The simulation of the elastodynamics equations at high frequency suffers from the well-known pollution effect. We present a Petrov–Galerkin multiscale sub-grid correction method that remains pollution-free in natural resolution and oversampling regimes. This is accomplished by generating corrections to coarse-grid spaces with supports determined by oversampling lengths related to the log ⁡ ( k ) \log(k) , 𝑘 being the wave number. Key to this method are polynomial-in-𝑘 bounds for stability constants and related inf-sup constants. To this end, we establish polynomial-in-𝑘 bounds for the elastodynamics stability constants in general Lipschitz domains with radiation boundary conditions in R 3 \mathbb{R}^{3} . Previous methods relied on variational techniques, Rellich identities, and geometric constraints. In the context of elastodynamics, these suffer from the need to hypothesize a Korn’s inequality on the boundary. The methods in this work are based on boundary integral operators and estimation of Green’s function’s derivatives dependence on 𝑘 and do not require this extra hypothesis. We also implemented numerical examples in two and three dimensions to show the method eliminates pollution in the natural resolution and oversampling regimes, as well as performs well when compared to standard Lagrange finite elements.
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