scholarly journals Partial Regularity for Nonlinear Subelliptic Systems with Dini Continuous Coefficients in Heisenberg Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Jialin Wang ◽  
Pingzhou Hong ◽  
Dongni Liao ◽  
Zefeng Yu
Keyword(s):  
Weak Solution ◽  
Heisenberg Group ◽  
Partial Regularity ◽  
Harmonic Approximation ◽  
Growth Conditions ◽  
Heisenberg Groups ◽  
Hölder Exponent ◽  
Hölder Continuous ◽  
Horizontal Gradients ◽  
Controllable Growth

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.

scholarly journals Regularity for sub-elliptic systems with VMO-coefficients in the Heisenberg group: the sub-quadratic structure case

2020 ◽  
Vol 10 (1) ◽  
pp. 420-449
Author(s):  
Jialin Wang ◽  
Maochun Zhu ◽  
Shujin Gao ◽  
Dongni Liao
Keyword(s):  
Heisenberg Group ◽  
Harmonic Approximation ◽  
Type Inequality ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Quadratic Growth ◽  
Primary Model ◽  
Vmo Coefficients ◽  
Controllable Growth ◽  
Vector Valued

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.

scholarly journals Partial Hölder continuity for nonlinear sub-elliptic systems with VMO-coefficients in the Heisenberg group

2018 ◽  
Vol 7 (1) ◽  
pp. 97-116 ◽  
Author(s):  
Jialin Wang ◽  
Juan J. Manfredi
Keyword(s):  
Heisenberg Group ◽  
Weak Solutions ◽  
Hölder Continuity ◽  
Harmonic Approximation ◽  
Elliptic Systems ◽  
Natural Growth ◽  
Holder Continuity ◽  
Model Case ◽  
Horizontal Gradients ◽  
Vmo Coefficients

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}.

scholarly journals Partial regularity up to the boundary for solutions of subquadratic elliptic systems

2018 ◽  
Vol 7 (4) ◽  
pp. 469-483 ◽  
Author(s):  
Zhong Tan ◽  
Yanzhen Wang ◽  
Shuhong Chen
Keyword(s):  
Partial Regularity ◽  
Direct Method ◽  
Elliptic Systems ◽  
Divergence Form ◽  
Singular Set ◽  
Nonlinear Elliptic Systems ◽  
Hölder Continuous ◽  
Nonlinear Elliptic ◽  
Controllable Growth Condition ◽  
Controllable Growth

AbstractIn this paper, we are concerned with the nonlinear elliptic systems in divergence form under controllable growth condition. We prove that the weak solution u is locally Hölder continuous besides a singular set by using the direct method and classical Morrey-type estimates. Here the Hausdorff dimension of the singular set is less than {n-p}. This result not only holds in the interior, but also holds up to the boundary.

scholarly journals Estimates for Operators Related to the Sub-Laplacian with Drift in Heisenberg Groups

2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Hong-Quan Li ◽  
Peter Sjögren
Keyword(s):  
Heisenberg Group ◽  
Weak Type ◽  
Riesz Transforms ◽  
Heisenberg Groups ◽  
Related Measure

AbstractIn the Heisenberg group of dimension $$2n+1$$ 2 n + 1 , we consider the sub-Laplacian with a drift in the horizontal coordinates. There is a related measure for which this operator is symmetric. The corresponding Riesz transforms are known to be $$L^p$$ L p bounded with respect to this measure. We prove that the Riesz transforms of order 1 are also of weak type (1, 1), and that this is false for order 3 and above. Further, we consider the related maximal Littlewood–Paley–Stein operators and prove the weak type (1, 1) for those of order 1 and disprove it for higher orders.

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