Examples of area-minimizing surfaces in the sub-Riemannian Heisenberg group $${\mathbb{H}^1}$$ with low regularity

2008 ◽  
Vol 34 (2) ◽  
pp. 179-192 ◽  
Author(s):  
Manuel Ritoré
Keyword(s):  
Heisenberg Group ◽  
Low Regularity ◽  
Area Minimizing

scholarly journals Area-minimizing cones in the Heisenberg group H

2021 ◽  
Vol 46 (2) ◽  
pp. 945-956 ◽  
Author(s):  
Sebastiano Nicolussi Golo ◽  
Manuel Ritoré
Keyword(s):  
Heisenberg Group ◽  
Area Minimizing

Weighted estimates for commutators of Hausdorff operators on the Heisenberg group

2020 ◽  
pp. 39-62
Author(s):  
Nguyen Minh Chuong ◽  
◽  
Dao Van Duong ◽  
Nguyen Duc Duyet ◽  
◽  
...  
Keyword(s):  
Heisenberg Group ◽  
Weighted Estimates ◽  
Hausdorff Operators

The horofunction boundary of the Heisenberg group

2009 ◽  
Vol 242 (2) ◽  
pp. 299-310 ◽  
Author(s):  
Tom Klein ◽  
Andrew Nicas
Keyword(s):  
Heisenberg Group

scholarly journals Finite-dimensional boundary uniform stabilization of the Boussinesq system in Besov spaces by critical use of Carleman estimate-based inverse theory

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Irena Lasiecka ◽  
Buddhika Priyasad ◽  
Roberto Triggiani
Keyword(s):  
Besov Space ◽  
Internal Control ◽  
Initial Conditions ◽  
Fluid Velocity ◽  
Critical Role ◽  
Boussinesq System ◽  
Heat Equations ◽  
Fluid Equation ◽  
Low Regularity ◽  
Finite Dimensional

Abstract We consider the 𝑑-dimensional Boussinesq system defined on a sufficiently smooth bounded domain and subject to a pair { v , u } \{v,\boldsymbol{u}\} of controls localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . Here, 𝑣 is a scalar Dirichlet boundary control for the thermal equation, acting on an arbitrarily small connected portion Γ ~ \widetilde{\Gamma} of the boundary Γ = ∂ ⁡ Ω \Gamma=\partial\Omega . Instead, 𝒖 is a 𝑑-dimensional internal control for the fluid equation acting on an arbitrarily small collar 𝜔 supported by Γ ~ \widetilde{\Gamma} . The initial conditions for both fluid and heat equations are taken of low regularity. We then seek to uniformly stabilize such Boussinesq system in the vicinity of an unstable equilibrium pair, in the critical setting of correspondingly low regularity spaces, by means of an explicitly constructed, finite-dimensional feedback control pair { v , u } \{v,\boldsymbol{u}\} localized on { Γ ~ , ω } \{\widetilde{\Gamma},\omega\} . In addition, they will be minimal in number and of reduced dimension; more precisely, 𝒖 will be of dimension ( d - 1 ) (d-1) , to include necessarily its 𝑑-th component, and 𝑣 will be of dimension 1. The resulting space of well-posedness and stabilization is a suitable, tight Besov space for the fluid velocity component (close to L 3 ⁢ ( Ω ) \boldsymbol{L}^{3}(\Omega) for d = 3 d=3 ) and a corresponding Besov space for the thermal component, q > d q>d . Unique continuation inverse theorems for suitably over-determined adjoint static problems play a critical role in the constructive solution. Their proof rests on Carleman-type estimates, a topic pioneered by M. V. Klibanov since the early 80s.

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