Large time behavior for the heat equation on Carnot groups

通过解析延拓方法构造卡诺群中亚拉普拉斯算子的分数次幂

The fractional powers of the sub-Laplacian in Carnot groups through an analytic continuation

本文通过解析延拓构造卡诺群中亚拉普拉斯算子的分数次幂，刻画海森堡群中该算子的幂次并计算热核k阶矩，为相关分析提供新方法与结果。

A Bourgain-Brezis-Mironescu-Dávila theorem in Carnot groups of step two

In this note we prove the following theorem in any Carnot group of step two $\mathbb{G}$: \[ \underset{s\nearrow 1/2}{\lim} (1 - 2s) \mathfrak P_{H,s}(E) = \frac{4}{\sqrt π}\ \mathfrak P_H(E). \] Here, $\mathfrak P_H(E)$ represents the horizontal perimeter of a measurable set $E\subset \mathbb{G}$, whereas the nonlocal horizontal perimeter $\mathfrak P_{H,s}(E)$ is a heat based Besov seminorm. This result represents a dimensionless sub-Riemannian counterpart of a famous characterisation of Bourgain-Brezis-Mironescu and Dávila.

关于Carnot群中Alt-Caffarelli-Friedman单调性公式的一些反例

Some counterexamples to Alt-Caffarelli-Friedman monotonicity formulas in Carnot groups

本文在阶数大于1的Carnot群中研究Alt-Caffarelli-Friedman单调性公式，利用调和多项式缺乏正交性给出存在单调递增行为反例的充分条件，推广了已有结果。

Function Spaces via Fractional Poisson Kernel on Carnot Groups and Applications

We provide a new characterization of homogeneous Besov and Sobolev spaces in Carnot groups using the fractional heat kernel and Poisson kernel. We apply our results to study commutators involving fractional powers of the sub-Laplacian.

Asymptotic behavior of entire solutions for degenerate partial differential inequalities on Carnot-Carathéodory metric spaces and Liouville type results

This article is devoted to the study of the behavior of generalized entire solutions for a wide class of quasilinear degenerate inequalities modeled on the following prototype with p-Laplacian in the main part \begin{equation*} {\underset{m}{\overset{i=1}{\sum}}} X_i^*(|\mathbf{X}u|^{p-2} X_i u)\geq |u|^{q-2}u, \ \ x\in {\mathbb{R}}^{n},\ q>1,\ p>1, \end{equation*} where ${\mathbb{R}}^{n}$ is a Carnot-Carathéodory metric space, generated by the system of vector fields $\mathbf{X}=(X_1,X_2,..,X_m)$ and $X_i^*$ denotes the adjoint of $X_i$ with respect to Lebesgue measure. For the case where $p$…

Reverse Poincare inequalities, Isoperimetry, and Riesz transforms in Carnot groups

We prove an optimal reverse Poincaré inequality for the heat semigroup generated by the sub-Laplacian on a Carnot group of any step. As an application we give new proofs of the isoperimetric inequality and of the boundedness of the Riesz transform in Carnot groups.

Carnot群中分数热含量的小时间渐进行为

Fractional heat content asymptotics for Carnot groups

研究Carnot群中C²非特征领域的小时间渐进行为，通过分数亚拉普拉斯算子定义分数热含量，并证明其渐近率函数与欧几里得情况一致。

Uniform Gaussian bounds for subelliptic heat kernels and an application to the total variation flow of graphs over Carnot groups

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…

Alt-Caffarelli-Friedman单调性公式与Carnot群中的均值性质及其应用

Alt-Caffarelli-Friedman monotonicity formula and mean value properties in Carnot groups with applications

本文提出一种基于梯度范数函数均值单调递增性质的新方法来处理Alt-Caffarelli-Friedman单调性公式，并证明该论证在Carnot群的一般框架下成立。

The asymptotic $p$-Poisson equation as $p \to \infty$ in Carnot-Carathéodory spaces

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.

Asymptotic behaviours in Fractional Orlicz-Sobolev spaces on Carnot groups

In this article we define a class of fractional Orlicz-Sobolev spaces on Carnot groups and, in the spirit of the celebrated results of Bourgain-Brezis-Mironescu and of Maz'ya-Shaposhnikova, we study the asymptotic behavior of the Orlicz functionals when the fractional parameter goes to $1$ and $0$.

A note on monotonicity and Bochner formulas in Carnot groups

In this note we prove two monotonicity formulas for solutions of $Δ_H f = c$ and $Δ_H f - \p_t f = c$ in Carnot groups. Such formulas involve the right-invariant \emph{carré du champ} of a function and they are false for the left-invariant one. The main results, Theorems 1.1 and 1.2, display a resemblance with two deep monotonicity formulas respectively due to Alt-Caffarelli-Friedman for the standard Laplacian, and to Caffarelli for the heat equation. In connection with this aspect we ask the question whether an "almost monotoniccity" formula be possible. In the last section we discuss the fai…

Sub-Riemannian heat kernels and mean curvature flow of graphs

We introduce a sub-Riemannian analogue of the Bence-Merriman-Osher diffusion driven algorithm and show that it leads to weak solutions of the horizontal mean curvature flow of graphs over sub-Riemannian Carnot groups. The proof follows the nonlinear semi-group theory approach originally introduced by L. C. Evans in the Euclidean setting and is based on new results on the relation between sub-Riemannian heat flows of characteristic functions of subgraphs and the horizontal mean curvature of the corresponding graphs.

Poincare inequality and the uniqueness of solutions for the heat equation associated with subelliptic diffusion operators

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.