For any Carnot group $\bf G$ and a bounded domain $Ω\subset \bf G$, we prove that viscosity solutions in $C(\bar\Om)$ of the fully nonlinear subelliptic equation $F(u,\nabla_h u, \nabla^2_h u)=0$ are unique when $F\in C(R\times R^m\times {\Cal S}(m))$ satisfies (i) $F$ is degenerate subelliptic and decreasing in $u$ or (ii) $F$ is uniformly subelliptic and nonincreasing in $u$. This extends Jensen's uniqueness theorem from the Euclidean space to the sub-Riemannian setting of the Carnot group.
Given a Carnot-Carathéodory space $\Om \sub \R^n$ with associated vector fields $X=\{X_1,...,X_m\}$, we derive the subelliptic $\infty$-Laplace system for mappings $u : \Om \larrow \R^N$, which reads \[ \label{1} \De^X_\infty u \, :=\, \Big(Xu \ot Xu + \|Xu\|^2 [Xu]^\bot \ot I \Big) : XX u\, = \, 0 \tag{1} \] in the limit of the subelliptic $p$-Laplacian as $p\ri \infty$. Here $Xu$ is the horizontal gradient and $[Xu]^\bot$ is the projection on its nullspace. Next, we identify the Variational Principle characterizing \eqref{1}, which is the "Euler-Lagrange PDE" of the supremal functional \[ \l…
We present two comparison principles for viscosity sub- and supersolutions of Monge-Ampere-type equations associated to a family of vector fields. In particular, we obtain the uniqueness of a viscosity solution to the Dirichlet problem for the equation of prescribed horizontal Gauss curvature in a Carnot group.
In this paper we prove an invariant Harnack inequality on Carnot-Carathéodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups.
The aim of this paper is to prove $Γ^{1,α}$ Schauder estimates near a $C^{1,α}$ non-characteristic portion of the boundary for $Γ^{0, α}$ perturbations of horizontal Laplaceans in Carnot groups. This situation of minimally smooth domains presents itself naturally in the study of subelliptic free boundary problems of obstacle type, see [15].
We prove surface and volume mean value formulas for classical solutions to uniformly elliptic equations in divergence form with Hölder continuous coefficients. The kernels appearing in the integrals are supported on the level and superlevel sets of the fundamental solution relevant the adjoint differential operator. We then extend the aforementioned formulas to some subelliptic operators on Carnot groups. In this case we rely on the theory of finite perimeters on stratified Lie groups.
This article is concerned with a class of elliptic equations on Carnot groups depending of one real positive parameter and involving a critical nonlinearity. As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic critical equation defined on a smooth and bounded domain $D$ of the {Heisenberg group} $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. Our approach is based on pure variational methods and locally sequentially weakly lower semicontinuous arguments.
In this paper we study the asymptotic behavior of solutions to the subelliptic $p$-Poisson equation as $p\to +\infty$ in Carnot Carathéodory spaces. In particular, introducing a suitable notion of differentiability, we extend the celebrated result of Bhattacharya, DiBenedetto and Manfredi [Rend. Sem. Mat. Univ. Politec. Torino, 1989, Special Issue, 15-68] and we prove that limits of such solutions solve in the sense of viscosity a hybrid first and second order PDE involving the $\infty-$Laplacian and the Eikonal equation.
In this paper we study heat kernels associated to a Carnot group $G$, endowed with a family of collapsing left-invariant Riemannian metrics $σ_\e$ which converge in the Gromov-Hausdorff sense to a sub-Riemannian structure on $G$ as $\e\to 0$. The main new contribution are Gaussian-type bounds on the heat kernel for the $σ_\e$ metrics which are stable as $\e\to 0$ and extend the previous time-independent estimates in \cite{CiMa-F}. As an application we study well posedness of the total variation flow of graph surfaces over a bounded domain in $(G,\s_\e)$. We establish interior and boundary grad…
Given a Carnot-Carathéodory metric space $(R^n, d_{\hbox{cc}})$ generated by vector fields $\{X_i\}_{i=1}^m$ satisfying Hörmander's condition, we prove in theorem A that any absolute minimizer $u\in W^{1,\infty}_{\hbox{cc}}(\Om)$ to $F(v,\Om)=\sup_{x\in\Om}f(x,Xv(x))$ is a viscosity solution to the Aronsson equation (1.6), under suitable conditions on $f$. In particular, any AMLE is a viscosity solution to the subelliptic $\infty$-Laplacian equation (1.7). If the Carnot-Carathédory space is a Carnot group ${\bf G}$ and $f$ is independent of $x$-variable, we establish in theorem C the uniquness…
We introduce a new notion of Almgren's frequency which is adapted to solutions of a sub-Laplacian (harmonic functions) on a Carnot group of arbitrary step $\bG$. With this notion we investigate some new functionals associated with the frequency, and obtain monotonicity formulas for the relevant harmonic functions, or for the solutions of a closely connected class of degenerate second order operators of Baouendi type. The results proved in this paper provide some new insight into the deep link existing between the growth properties of the frequency, and the local and global structure of the rel…
This article concerns a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity (for the critical case we refer to G. Molica Bisci and D. Repovš, Yamabe-type equations on Carnot groups, Potential Anal. 46:2 (2017), 369-383; arXiv:1705.10100 [math.AP]). As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic equation defined on a smooth and bounded domain $D$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. The main approach is based on variational me…
In the sub-Riemannian setting of Carnot groups, this work investigates a-priori estimates and Liouville type theorems for solutions of coercive, quasilinear differential inequalities of the type $$ Δ_{\mathbb{G}}^φu \ge b(x) f(u) l(|\nabla u|). $$ Prototype examples of $Δ_{\mathbb{G}}^φ$ are the (subelliptic) $p$-Laplacian and the mean curvature operator. The main novelty of the present paper is that we allow a dependence on the gradient $l(t)$ that can vanish both as $t \rightarrow 0^+$ and as $t \rightarrow +\infty$. Our results improve on the recent literature and, by means of suitable coun…
In this paper we study global Poincare inequalities on balls in a large class of sub-Riemannian manifolds satisfying the generalized curvature dimension inequality introduced by F.Baudoin and N.Garofalo. As a corollary, we prove the uniqueness of solutions for the subelliptic heat equation. Our results apply in particular to CR Sasakian manifolds with Tanaka-Webster-Ricci curvature bounded from below and Carnot groups of step two.
We establish precise upper and lower bounds for the subelliptic heat kernel on nilpotent Lie groups $G$ of H-type. Specifically, we show that there exist positive constants $C_1$, $C_2$ and a polynomial correction function $Q_t$ on $G$ such that $$C_1 Q_t e^{-\frac{d^2}{4t}} \le p_t \le C_2 Q_t e^{-\frac{d^2}{4t}}$$ where $p_t$ is the heat kernel, and $d$ the Carnot-Carathéodory distance on $G$. We also obtain similar bounds on the norm of its subelliptic gradient $|\nabla p_t|$. Along the way, we record explicit formulas for the distance function $d$ and the subriemannian geodesics of H-type …
We study the Dirichlet problem for subelliptic partial differential equations of Monge-Ampere type involving the derivates with respect to a family X of vector fields of Carnot type. The main result is a comparison principle among viscosity subsolutions, convex with respect to X, and viscosity supersolutions (in a weaker sense than usual), which implies the uniqueness of solution to the Dirichlet problem. Its assumptions include the equation of prescribed horizontal Gauss curvature in Carnot groups. By Perron method we also prove the existence of a solution either under a growth condition of t…
We introduce a notion of subunit vector field for fully nonlinear degenerate elliptic equations. We prove that an interior maximum of a viscosity subsolution of such an equation propagates along the trajectories of subunit vector fields. This implies strong maximum and minimum principles when the operator has a family of subunit vector fields satisfying the Hörmander condition. In particular these results hold for a large class of nonlinear subelliptic PDEs in Carnot groups. We prove also a strong comparison principle for degenerate elliptic equations that can be written in Hamilton-Jacobi-Bel…
