Estimates of Dirichlet eigenvalues for fractional sub-Laplacian operators on Carnot groups

发布时间：2023-09-08 作者：浏览次数：

Speaker:	陈洪葛	DateTime:	2023年9月11日（周一）上午10:30-11:30
Brief Introduction to Speaker: 陈洪葛主要研究一般非等度正则次黎曼流形上退化Hrmander算子的特征值问题，给出了特征值Weyl型渐近式成立的充要条件，从而证明了这类次黎曼流形的非各向同性维数是几何谱不变量，同时还首次研究了系数非光滑的退化椭圆算子特征值的精确估计。其多项科研成果分别在Proc. Lond. Math. Soc., J. Math. Pures Appl.和Calc. Var. Partial Differential Equations等国际权威数学期刊上发表。		Place:	6号楼2楼报告厅
Abstract:In this talk, we shall present some recent results on the Dirichlet eigenvalue problem offractional sub-Laplacian $(-\triangle_{\mathbb{G}})^{s}$ on homogeneous Carnot group $\mathbb{G}=(\mathbb{R}^{n},\circ)$. Let $\Omega$ be a bounded open domain of $\mathbb{G}$ and denote by $\lambda_{k}$ the $k$-th Dirichlet eigenvalue of the fractional sub-Laplacian operator $(-\triangle_{\mathbb{G}})^{s}$ on $\Omega$. We give explicit estimates for the trace of the Dirichlet heat kernel of $(-\triangle_{\mathbb{G}})^{s}$ via a comparison of heat kernels. Based on these estimates, we obtain an explicit lower bound estimate for $\lambda_k$, which exhibits the optimal growth order of $k$. Then, we establish the Weyl's law for the spectral counting function $N(\lambda)$. In particular, under a certain geometric condition of $\Omega$, we also provide the reminder term estimate of the trace of the Dirichlet heat kernel and an explicit upper bound of $\lambda_k$ with optimal growth order of $k$. Furt...

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