Relevant bibliographies by topics / Rellich inequality

Academic literature on the topic 'Rellich inequality'

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Published: 4 June 2021
Last updated: 28 January 2023

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Journal articles on the topic "Rellich inequality"

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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Rellich inequality"

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Cabral, Rodrigo Augusto Higo Mafra. "O teorema espectral e a propriedade de \"self-adjointness\" para alguns operadores de Schrödinger." Universidade de São Paulo, 2014. http://www.teses.usp.br/teses/disponiveis/45/45132/tde-07032015-154510/.

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Neste texto são demonstrados, a partir do ponto de vista da teoria dos espaços de Hilbert e da teoria das C*-álgebras, teoremas relacionados a operadores auto-adjuntos em espaços de Hilbert, entre os quais estão o Teorema Espectral, o teorema de Kato-Rellich e a desigualdade de Kato. Também são dadas aplicações destes teoremas a alguns operadores de Schrödinger provenientes da Física-Matemática.
In this text we prove, within the Hilbert spaces theory and C*-algebras points of view, some theorems which are related to self-adjoint operators acting on Hilbert spaces, among which are the Spectral Theorem, the Kato-Rellich theorem and Kato\'s inequality. Also, some applications to Schrödinger operators coming from the Mathematical-Physics context are given.
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Axelsson, Andreas, and kax74@yahoo se. "Transmission problems for Dirac's and Maxwell's equations with Lipschitz interfaces." The Australian National University. School of Mathematical Sciences, 2002. http://thesis.anu.edu.au./public/adt-ANU20050106.093019.

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The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface,in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.
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Axelsson, Andreas. "Transmission problems for Dirac's and Maxwell's equations with Lipschitz interfaces." Phd thesis, 2002. http://hdl.handle.net/1885/46056.

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The aim of this thesis is to give a mathematical framework for scattering of electromagnetic waves by rough surfaces. We prove that the Maxwell transmission problem with a weakly Lipschitz interface, in finite energy norms, is well posed in Fredholm sense for real frequencies. Furthermore, we give precise conditions on the material constants ε, μ and σ and the frequency ω when this transmission problem is well posed. To solve the Maxwell transmission problem, we embed Maxwell’s equations in an elliptic Dirac equation. We develop a new boundary integral method to solve the Dirac transmission problem. This method uses a boundary integral operator, the rotation operator, which factorises the double layer potential operator. We prove spectral estimates for this rotation operator in finite energy norms using Hodge decompositions on weakly Lipschitz domains. To ensure that solutions to the Dirac transmission problem indeed solve Maxwell’s equations, we introduce an exterior/interior derivative operator acting in the trace space. By showing that this operator commutes with the two basic reflection operators, we are able to prove that the Maxwell transmission problem is well posed. We also prove well-posedness for a class of oblique Dirac transmission problems with a strongly Lipschitz interface, in the L_2 space on the interface. This is shown by employing the Rellich technique, which gives angular spectral estimates on the rotation operator.
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Book chapters on the topic "Rellich inequality"

1

Balinsky, Alexander A., W. Desmond Evans, and Roger T. Lewis. "The Rellich Inequality." In The Analysis and Geometry of Hardy's Inequality, 213–49. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-22870-9_6.

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Edmunds, David E., and W. Desmond Evans. "The Rellich Inequality." In Springer Monographs in Mathematics, 235–47. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-030-02125-2_10.

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