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

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Author: Grafiati

Published: 10 March 2023

Last updated: 11 March 2023

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1

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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2

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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3

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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4

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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5

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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6

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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7

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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8

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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9

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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10

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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11

Wang, Jialin, and Juan J. Manfredi. "Partial Hölder continuity for nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group." Advances in Nonlinear Analysis 7, no. 1 (February 1, 2018): 97–116. http://dx.doi.org/10.1515/anona-2015-0182.

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AbstractWe consider nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group and prove partial Hölder continuity results for weak solutions using a generalization of the technique of {\mathcal{A}}-harmonic approximation. The model case is the following non-degenerate p-sub-Laplace system with super-quadratic natural growth with respect to the horizontal gradients Xu:-\sum_{i=1}^{2n}X_{i}\bigl{(}a(\xi\/)(1+|Xu|^{2})^{{(p-2)/2}}X_{i}u^{\alpha}% \bigr{)}=f^{\alpha},\quad\alpha=1,2,\ldots,N,where {a(\xi\/)\in\mathrm{VMO}} and {2<p<\infty}.

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12

Zhang, Caifeng, and Lu Chen. "Concentration-Compactness Principle of Singular Trudinger--Moser Inequalities in ℝ n and n-Laplace Equations." Advanced Nonlinear Studies 18, no. 3 (August 1, 2018): 567–85. http://dx.doi.org/10.1515/ans-2017-6041.

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AbstractIn this paper, we use the rearrangement-free argument, in the spirit of the work by Li, Lu and Zhu [25], on the concentration-compactness principle on the Heisenberg group to establish a sharpened version of the singular Lions concentration-compactness principle for the Trudinger–Moser inequality in{\mathbb{R}^{n}}. Then we prove a compact embedding theorem, which states that{W^{1,n}(\mathbb{R}^{n})}is compactly embedded into{L^{p}(\mathbb{R}^{n},|x|^{-\beta}\,dx)}for{p\geq n}and{0<\beta<n}. As an application of the above results, we establish sufficient conditions for the existence of ground state solutions to the followingn-Laplace equation with critical nonlinearity:($*$){}\left\{\begin{aligned} &\displaystyle{-}\operatorname{div}(|\nabla u|^{n-2}% \nabla u)+V(x)|u|^{n-2}u=\frac{f(x,u)}{|x|^{\beta}},\\ &\displaystyle u\in W^{1,n}(\mathbb{R}^{n}),\quad u\geq 0,\end{aligned}\right.where{V(x)\geq c_{0}}for some positive constant{c_{0}}and{f(x,t)}behaves like{\exp(\alpha|t|^{\frac{n}{n-1}})}as{t\rightarrow+\infty}. This work improves substantially related results found in the literature.

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13

Ruzhansky, Michael, Niyaz Tokmagambetov, and Nurgissa Yessirkegenov. "Best constants in Sobolev and Gagliardo–Nirenberg inequalities on graded groups and ground states for higher order nonlinear subelliptic equations." Calculus of Variations and Partial Differential Equations 59, no. 5 (September 16, 2020). http://dx.doi.org/10.1007/s00526-020-01835-0.

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Abstract In this paper the dependence of the best constants in Sobolev and Gagliardo–Nirenberg inequalities on the precise form of the Sobolev space norm is investigated. The analysis is carried out on general graded Lie groups, thus including the cases of $$\mathbb {R}^n$$ R n , Heisenberg, and general stratified Lie groups, in all these cases the results being new. The Sobolev norms may be defined in terms of Rockland operators, i.e. the hypoelliptic homogeneous left-invariant differential operators on the group. The best constants are expressed in the variational form as well as in terms of the ground state solutions of the corresponding nonlinear subelliptic equations. The orders of these equations can be high depending on the Sobolev space order in the Sobolev or Gagliardo–Nirenberg inequalities, or may be fractional. Applications are obtained also to equations with lower order terms given by different hypoelliptic operators. Already in the case of $${\mathbb {R}}^n$$ R n , the obtained results extend the classical relations by Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) to a wide range of nonlinear elliptic equations of high orders with elliptic low order terms and a wide range of interpolation inequalities of Gagliardo–Nirenberg type. However, the proofs are different from those in Weinstein (Commun Math Phys 87(4):567–576 (1982/1983)) because of the impossibility of using the rearrangement inequalities already in the setting of the Heisenberg group. The considered class of graded groups is the most general class of nilpotent Lie groups where one can still consider hypoelliptic homogeneous invariant differential operators and the corresponding subelliptic differential equations.

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14

Palatucci, Giampiero, and Mirco Piccinini. "Nonlocal Harnack inequalities in the Heisenberg group." Calculus of Variations and Partial Differential Equations 61, no. 5 (July 26, 2022). http://dx.doi.org/10.1007/s00526-022-02301-9.

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AbstractWe deal with a wide class of nonlinear integro-differential problems in the Heisenberg-Weyl group $$\mathbb {H}^n$$ H n , whose prototype is the Dirichlet problem for the p-fractional subLaplace equation. These problems arise in many different contexts in quantum mechanics, in ferromagnetic analysis, in phase transition problems, in image segmentations models, and so on, when non-Euclidean geometry frameworks and nonlocal long-range interactions do naturally occur. We prove general Harnack inequalities for the related weak solutions. Also, in the case when the growth exponent is $$p=2$$ p = 2 , we investigate the asymptotic behavior of the fractional subLaplacian operator, and the robustness of the aforementioned Harnack estimates as the differentiability exponent s goes to 1.

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15

Abolarinwa, Abimbola. "Weighted Hardy and Rellich Types Inequalities on the Heisenberg Group with Sharp Constants." Journal of Nonlinear Mathematical Physics, January 11, 2023. http://dx.doi.org/10.1007/s44198-022-00105-1.

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AbstractThis paper aims at deriving some weighted Hardy type and Rellich type inequalities with sharp constants on the Heisenberg group. The improved versions of these inequalities are established as well. The technique adopted involve the application of some elementary vectorial inequalities and some properties of Heisenberg group.

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16

Wu, Duan, and Pengcheng Niu. "An oscillation theorem for the nonlinear degenerate elliptic equation in the Heisenberg group." Boundary Value Problems 2021, no. 1 (April 7, 2021). http://dx.doi.org/10.1186/s13661-021-01516-7.

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AbstractThe aim of this paper is to study the oscillation of solutions of the nonlinear degenerate elliptic equation in the Heisenberg group $H^{n}$ H n . We first derive a critical inequality in $H^{n}$ H n . Based on it, we establish a Picone-type differential inequality and a Sturm-type comparison principle. Then we obtain an oscillation theorem. Our result generalizes the related conclusions for the nonlinear elliptic equations in $R^{n}$ R n .

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17

Bardi, Martino, and Alessandro Goffi. "Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group." Mathematische Annalen, December 23, 2020. http://dx.doi.org/10.1007/s00208-020-02118-x.

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AbstractThis paper studies Liouville properties for viscosity sub- and supersolutions of fully nonlinear degenerate elliptic PDEs, under the main assumption that the operator has a family of generalized subunit vector fields that satisfy the Hörmander condition. A general set of sufficient conditions is given such that all subsolutions bounded above are constant; it includes the existence of a supersolution out of a big ball, that explodes at infinity. Therefore for a large class of operators the problem is reduced to finding such a Lyapunov-like function. This is done here for the vector fields that generate the Heisenberg group, giving explicit conditions on the sign and size of the first and zero-th order terms in the equation. The optimality of the conditions is shown via several examples. A sequel of this paper applies the methods to other Carnot groups and to Grushin geometries.

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18

Zhang, Junli, and Jialin Wang. "Partial regularity for a nonlinear discontinuous sub-elliptic system with drift on the Heisenberg group: the superquadratic case." Complex Variables and Elliptic Equations, December 2, 2022, 1–26. http://dx.doi.org/10.1080/17476933.2022.2152444.

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19

Jleli, Mohamed, Mokhtar Kirane, and Bessem Samet. "Liouville-type results for elliptic equations with advection and potential terms on the Heisenberg group." Discrete and Continuous Dynamical Systems - S, 2022, 0. http://dx.doi.org/10.3934/dcdss.2022171.

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<p style='text-indent:20px;'>We investigate nonlinear elliptic equations of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta_{H} u(\xi)+ A(\xi) \cdot \nabla_{H} u(\xi) = V(\xi)f(u),\quad \xi\in \mathbb{H}^n, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{H}^n = (\mathbb{R}^{2n+1},\circ) $\end{document}</tex-math></inline-formula> is the <inline-formula><tex-math id="M2">\begin{document}$ (2n+1) $\end{document}</tex-math></inline-formula>-dimensional Heisenberg group, <inline-formula><tex-math id="M3">\begin{document}$ \Delta_{H} $\end{document}</tex-math></inline-formula> is the Kohn-Laplacian operator, <inline-formula><tex-math id="M4">\begin{document}$ \nabla_{H} $\end{document}</tex-math></inline-formula> is the Heisenberg gradient, <inline-formula><tex-math id="M5">\begin{document}$ \cdot $\end{document}</tex-math></inline-formula> is the inner product in <inline-formula><tex-math id="M6">\begin{document}$ \mathbb{R}^{2n} $\end{document}</tex-math></inline-formula>, the advection term <inline-formula><tex-math id="M7">\begin{document}$ A: \mathbb{H}^n\to \mathbb{R}^{2n} $\end{document}</tex-math></inline-formula> is a <inline-formula><tex-math id="M8">\begin{document}$ C^1 $\end{document}</tex-math></inline-formula> vector field satisfying a certain decay condition, the potential function <inline-formula><tex-math id="M9">\begin{document}$ V: \mathbb{H}^n\to (0,\infty) $\end{document}</tex-math></inline-formula> is continuous, and the nonlinearity <inline-formula><tex-math id="M10">\begin{document}$ f(u) $\end{document}</tex-math></inline-formula> has the form <inline-formula><tex-math id="M11">\begin{document}$ -u^{-p} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ p>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ u>0 $\end{document}</tex-math></inline-formula>, or <inline-formula><tex-math id="M14">\begin{document}$ e^u $\end{document}</tex-math></inline-formula>. Namely, we establish Liouville-type results for the class of stable solutions to the considered problems. Next, some special cases of the potential function <inline-formula><tex-math id="M15">\begin{document}$ V $\end{document}</tex-math></inline-formula> are discussed.</p>

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