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Academic literature on the topic 'Nonlinear elliptic inequalities on the Heisenberg group'

Author: Grafiati
Published: 10 March 2023
Last updated: 11 March 2023

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Journal articles on the topic "Nonlinear elliptic inequalities on the Heisenberg group"

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Bordoni, Sara, Roberta Filippucci, and Patrizia Pucci. "Nonlinear elliptic inequalities with gradient terms on the Heisenberg group." Nonlinear Analysis: Theory, Methods & Applications 121 (July 2015): 262–79. http://dx.doi.org/10.1016/j.na.2015.02.012.

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Flynn, Joshua, Nguyen Lam, and Guozhen Lu. "Sharp Hardy Identities and Inequalities on Carnot Groups." Advanced Nonlinear Studies 21, no. 2 (March 12, 2021): 281–302. http://dx.doi.org/10.1515/ans-2021-2123.

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Abstract In this paper we establish general weighted Hardy identities for several subelliptic settings including Hardy identities on the Heisenberg group, Carnot groups with respect to a homogeneous gauge and Carnot–Carathéodory metric, general nilpotent groups, and certain families of Hörmander vector fields. We also introduce new weighted uncertainty principles in these settings. This is done by continuing the program initiated by [N. Lam, G. Lu and L. Zhang, Factorizations and Hardy’s-type identities and inequalities on upper half spaces, Calc. Var. Partial Differential Equations 58 2019, 6, Paper No. 183; N. Lam, G. Lu and L. Zhang, Geometric Hardy’s inequalities with general distance functions, J. Funct. Anal. 279 2020, 8, Article ID 108673] of using the Bessel pairs introduced by [N. Ghoussoub and A. Moradifam, Functional Inequalities: New Perspectives and New Applications, Math. Surveys Monogr. 187, American Mathematical Society, Providence, 2013] to obtain Hardy identities. Using these identities, we are able to improve significantly existing Hardy inequalities in the literature in the aforementioned subelliptic settings. In particular, we establish the Hardy identities and inequalities in the spirit of [H. Brezis and J. L. Vázquez, Blow-up solutions of some nonlinear elliptic problems, Rev. Mat. Univ. Complut. Madrid 10 1997, 443–469] and [H. Brezis and M. Marcus, Hardy’s inequalities revisited. Dedicated to Ennio De Giorgi, Ann. Sc. Norm. Super. Pisa Cl. Sci. (4) 25 1997, 1–2, 217–237] in these settings.
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Ruzhansky, Michael, Bolys Sabitbek, and Durvudkhan Suragan. "Geometric Hardy and Hardy–Sobolev inequalities on Heisenberg groups." Bulletin of Mathematical Sciences 10, no. 03 (July 4, 2020): 2050016. http://dx.doi.org/10.1142/s1664360720500162.

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In this paper, we present geometric Hardy inequalities for the sub-Laplacian in half-spaces of stratified groups. As a consequence, we obtain the following geometric Hardy inequality in a half-space of the Heisenberg group with a sharp constant: [Formula: see text] which solves a conjecture in the paper [S. Larson, Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domain in the Heisenberg group, Bull. Math. Sci. 6 (2016) 335–352]. Here, [Formula: see text] is the angle function. Also, we obtain a version of the Hardy–Sobolev inequality in a half-space of the Heisenberg group: [Formula: see text] where [Formula: see text] is the Euclidean distance to the boundary, [Formula: see text], and [Formula: see text]. For [Formula: see text], this gives the Hardy–Sobolev–Maz’ya inequality on the Heisenberg group.
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Wang, Jialin, Maochun Zhu, Shujin Gao, and Dongni Liao. "Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case." Advances in Nonlinear Analysis 10, no. 1 (August 7, 2020): 420–49. http://dx.doi.org/10.1515/anona-2020-0145.

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Abstract We consider nonlinear sub-elliptic systems with VMO-coefficients for the case 1 < p < 2 under controllable growth conditions, as well as natural growth conditions, respectively, in the Heisenberg group. On the basis of a generalization of the technique of 𝓐-harmonic approximation introduced by Duzaar-Grotowski-Kronz, and an appropriate Sobolev-Poincaré type inequality established in the Heisenberg group, we prove partial Hölder continuity results for vector-valued solutions of discontinuous sub-elliptic problems. The primary model covered by our analysis is the non-degenerate sub-elliptic p-Laplacian system with VMO-coefficients, involving sub-quadratic growth terms.
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Pucci, Patrizia, and Letizia Temperini. "Existence for (p, q) critical systems in the Heisenberg group." Advances in Nonlinear Analysis 9, no. 1 (November 7, 2019): 895–922. http://dx.doi.org/10.1515/anona-2020-0032.

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Abstract This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p, q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems.
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Du, Feng, Chuanxi Wu, Guanghan Li, and Changyu Xia. "Universal inequalities for eigenvalues of a system of sub-elliptic equations on Heisenberg group." Kodai Mathematical Journal 38, no. 2 (June 2015): 437–50. http://dx.doi.org/10.2996/kmj/1436403899.

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Larson, Simon. "Geometric Hardy inequalities for the sub-elliptic Laplacian on convex domains in the Heisenberg group." Bulletin of Mathematical Sciences 6, no. 3 (April 28, 2016): 335–52. http://dx.doi.org/10.1007/s13373-016-0083-4.

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Wang, YanYan Li and Bo. "Comparison Principles for Some Fully Nonlinear Sub-Elliptic Equations on the Heisenberg Group." Analysis in Theory and Applications 35, no. 3 (June 2019): 312–34. http://dx.doi.org/10.4208/ata.oa-0010.

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Wang, Jialin, Pingzhou Hong, Dongni Liao, and Zefeng Yu. "Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups." Abstract and Applied Analysis 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/950134.

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This paper is concerned with partial regularity to nonlinear subelliptic systems with Dini continuous coefficients under quadratic controllable growth conditions in the Heisenberg groupℍn. Based on a generalization of the technique of𝒜-harmonic approximation introduced by Duzaar and Steffen, partial regularity to the sub-elliptic system is established in the Heisenberg group. Our result is optimal in the sense that in the case of Hölder continuous coefficients we establish the optimal Hölder exponent for the horizontal gradients of the weak solution on its regular set.
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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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Dissertations / Theses on the topic "Nonlinear elliptic inequalities on the Heisenberg group"

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BORDONI, SARA. "Nonlinear elliptic problems in the Heisenberg group." Doctoral thesis, 2018. http://hdl.handle.net/2158/1121183.

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The aim of this Ph.D. thesis is to present new results concerning the study of nonlinear elliptic problems in the context of the Heisenberg group. We deal with different problems, but the common thread consists in extending to a more general setting, the Heisenberg group, results proved in the Euclidean case. This generalization process in the Heisenberg framework implies a series of technical difficulties, that force the use of new key theorems.
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Book chapters on the topic "Nonlinear elliptic inequalities on the Heisenberg group"

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Chen, Jianqing, and Eugénio M. Rocha. "A Class of Sub-elliptic Equations on the Heisenberg Group and Related Interpolation Inequalities." In Advances in Harmonic Analysis and Operator Theory, 123–37. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0516-2_7.

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