In this paper, we establish a scale invariant Harnack inequality for the fractional powers of par... more In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators [Formula: see text]. Our focus is on non-Euclidean situations.
Nonlinear Differential Equations and Applications NoDEA, 2018
We establish the existence and uniqueness of variational solution to the nonlinear Neumann bounda... more We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the pth-sub-Laplacian associated to a system of Hörmander vector fields.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
We announce the following result: if a domain Ω on the Heisenberg group ℍ n satisfies the (ε,δ) c... more We announce the following result: if a domain Ω on the Heisenberg group ℍ n satisfies the (ε,δ) condition, then there is a bounded extension operator ℰ from ℒ k,p (Ω) into ℒ k,p (ℍ n ) where 1≤k, 1≤p≤∞.
Let G be a group of Heisenberg type with homogeneous dimension Q. For every 0 < ε < Q we co... more Let G be a group of Heisenberg type with homogeneous dimension Q. For every 0 < ε < Q we construct a non-divergence form operator L-epsilon and a non-trivial solution u(epsilon) is an element of L-2,L-Q-epsilon (Omega) boolean AND C((&UOmega;) over bar) to the Dirichlet problem: Lu = 0 in Omega, u = 0 on partial derivativeOmega. This non-uniqueness result shows the impossibility of controlling the maximum of u with an L-p norm of Lu when p < Q. Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as [GRAPHICS] where m is the dimension of the horizontal layer of the Lie algebra and (u,(ij)) is the symmetrized horizontal Hessian of u.
Contemporary Mathematics Volume 277, 2001 Sub-elliptic Besov spaces and the characterization of t... more Contemporary Mathematics Volume 277, 2001 Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu 1. General theory In classical analysis Besov spaces play a fundamental role in the ...
We prove that if a domain $\Omega$ on the Heisenberg group $\IH\sp{n}$ satisfies the $(\epsilon ,... more We prove that if a domain $\Omega$ on the Heisenberg group $\IH\sp{n}$ satisfies the $(\epsilon ,\delta)$ condition then there is a linear bounded extension operator ${\cal E}$ from ${\cal L}\sp{k,p}(\Omega )$ into ${\cal L}\sp{k,p}(\IH\sp{n})$ where $1\le k,\ 1\le p\le\infty$. ^
We provide a solution to the isoperimetric problem in the Heisenberg group H n when the competing... more We provide a solution to the isoperimetric problem in the Heisenberg group H n when the competing sets belong to a restricted class of C 2 graphs. Within this restricted class we characterize the isoperimetric profiles as the bubble sets (1.5) (modulo nonisotropic dilations and left-translations). We also compute the isoperimetric constant. Contents 1. Introduction 1 2. Isoperimetric inequalities in Carnot groups 7 3. Partial solution of the isoperimetric problem in H n 13 References 35
Communications on Pure and Applied Mathematics, 1996
... Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of mi... more ... Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Nicola Garofalo,; Duy-Minh Nhieu. Article first published online: 6 DEC 1998. ... Correspondence: Nicola Garofalo, Purdue University. Publication History. ...
MINIMAL SURFACES, SURFACES OF CONSTANT MEAN CURVATURE AND ISOPERIMETRY IN SUB-RIEMANNIAN GROUPS D... more MINIMAL SURFACES, SURFACES OF CONSTANT MEAN CURVATURE AND ISOPERIMETRY IN SUB-RIEMANNIAN GROUPS DONATELLA DANIELLI, NICOLA GAROFALO, AND DUY-MINH NHIEU ... 2 DONATELLA DANIELLI, NICOLA GAROFALO, AND DUY-MINH NHIEU ...
In this paper, we establish a scale invariant Harnack inequality for the fractional powers of par... more In this paper, we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators [Formula: see text], [Formula: see text], where [Formula: see text] is the infinitesimal generator of a class of symmetric semigroups. As a by-product, we also obtain a similar result for the nonlocal operators [Formula: see text]. Our focus is on non-Euclidean situations.
Nonlinear Differential Equations and Applications NoDEA, 2018
We establish the existence and uniqueness of variational solution to the nonlinear Neumann bounda... more We establish the existence and uniqueness of variational solution to the nonlinear Neumann boundary problem for the pth-sub-Laplacian associated to a system of Hörmander vector fields.
Comptes Rendus de l Académie des Sciences - Series I - Mathematics
We announce the following result: if a domain Ω on the Heisenberg group ℍ n satisfies the (ε,δ) c... more We announce the following result: if a domain Ω on the Heisenberg group ℍ n satisfies the (ε,δ) condition, then there is a bounded extension operator ℰ from ℒ k,p (Ω) into ℒ k,p (ℍ n ) where 1≤k, 1≤p≤∞.
Let G be a group of Heisenberg type with homogeneous dimension Q. For every 0 < ε < Q we co... more Let G be a group of Heisenberg type with homogeneous dimension Q. For every 0 < ε < Q we construct a non-divergence form operator L-epsilon and a non-trivial solution u(epsilon) is an element of L-2,L-Q-epsilon (Omega) boolean AND C((&UOmega;) over bar) to the Dirichlet problem: Lu = 0 in Omega, u = 0 on partial derivativeOmega. This non-uniqueness result shows the impossibility of controlling the maximum of u with an L-p norm of Lu when p < Q. Another consequence is the impossiblity of an Alexandrov-Bakelman type estimate such as [GRAPHICS] where m is the dimension of the horizontal layer of the Lie algebra and (u,(ij)) is the symmetrized horizontal Hessian of u.
Contemporary Mathematics Volume 277, 2001 Sub-elliptic Besov spaces and the characterization of t... more Contemporary Mathematics Volume 277, 2001 Sub-elliptic Besov spaces and the characterization of traces on lower dimensional manifolds Donatella Danielli, Nicola Garofalo, and Duy-Minh Nhieu 1. General theory In classical analysis Besov spaces play a fundamental role in the ...
We prove that if a domain $\Omega$ on the Heisenberg group $\IH\sp{n}$ satisfies the $(\epsilon ,... more We prove that if a domain $\Omega$ on the Heisenberg group $\IH\sp{n}$ satisfies the $(\epsilon ,\delta)$ condition then there is a linear bounded extension operator ${\cal E}$ from ${\cal L}\sp{k,p}(\Omega )$ into ${\cal L}\sp{k,p}(\IH\sp{n})$ where $1\le k,\ 1\le p\le\infty$. ^
We provide a solution to the isoperimetric problem in the Heisenberg group H n when the competing... more We provide a solution to the isoperimetric problem in the Heisenberg group H n when the competing sets belong to a restricted class of C 2 graphs. Within this restricted class we characterize the isoperimetric profiles as the bubble sets (1.5) (modulo nonisotropic dilations and left-translations). We also compute the isoperimetric constant. Contents 1. Introduction 1 2. Isoperimetric inequalities in Carnot groups 7 3. Partial solution of the isoperimetric problem in H n 13 References 35
Communications on Pure and Applied Mathematics, 1996
... Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of mi... more ... Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of minimal surfaces. Nicola Garofalo,; Duy-Minh Nhieu. Article first published online: 6 DEC 1998. ... Correspondence: Nicola Garofalo, Purdue University. Publication History. ...
MINIMAL SURFACES, SURFACES OF CONSTANT MEAN CURVATURE AND ISOPERIMETRY IN SUB-RIEMANNIAN GROUPS D... more MINIMAL SURFACES, SURFACES OF CONSTANT MEAN CURVATURE AND ISOPERIMETRY IN SUB-RIEMANNIAN GROUPS DONATELLA DANIELLI, NICOLA GAROFALO, AND DUY-MINH NHIEU ... 2 DONATELLA DANIELLI, NICOLA GAROFALO, AND DUY-MINH NHIEU ...
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