论文标题
Dirichlet特征值的渐近行为，用于同质Hörmander操作员和代数几何方法
Asymptotic behaviour of Dirichlet eigenvalues for homogeneous Hörmander operators and algebraic geometry approach
论文作者
论文摘要
我们研究同质性Hörmander操作员的差异特征问题$ \ triangle_ {x} = \ sum_ {j = 1}^{m}^{m} x_ {j}^{j}^{2}^{2} $ in BOINDED OPINAIN包含一个原点，其中$ x_1，x_1，x_1，x_1，x_2，x_2，\ ldots，\ ldots，\ ldots，\ ldots，x___________。 $ \ mathbb {r}^n $满足了Hörmander的状况，并且相对于非异端扩张家庭具有合适的同质性属性。假设$ω$是包含原点的$ \ mathbb {r}^n $中的一个开放界域。我们使用DIRICHLET形式研究热半群和亚小节热核。然后，通过利用亚细热核估计值，代数几何形状的奇异性分辨率以及采用涉及凸几何形状的一些精制分析，我们建立了明显的渐近行为$λ_K\ of K^\ of K^{\ of k^{\ frac {2} {2} {q_0} {q_0} {Q_0} {\ ln k)^{-\frac{2d_0}{Q_0}}$ as $k \to +\infty$, where $λ_k$ denotes the $k$-th Dirichlet eigenvalue of $\triangle_{X}$ on $Ω$, $Q_0$ is a positive rational number, and $d_0$ is a non-negative integer.此外，我们提供索引$ q_0 $的最佳界限，该界限取决于与矢量字段$ x_1，x_2，\ ldots，x_m $相关的均质维度。
We study the Dirichlet eigenvalue problem of homogeneous Hörmander operators $\triangle_{X}=\sum_{j=1}^{m}X_{j}^{2}$ on a bounded open domain containing the origin, where $X_1, X_2, \ldots, X_m$ are linearly independent smooth vector fields in $\mathbb{R}^n$ satisfying Hörmander's condition and a suitable homogeneity property with respect to a family of non-isotropic dilations. Suppose that $Ω$ is an open bounded domain in $\mathbb{R}^n$ containing the origin. We use the Dirichlet form to study heat semigroups and subelliptic heat kernels. Then, by utilizing subelliptic heat kernel estimates, the resolution of singularities in algebraic geometry, and employing some refined analysis involving convex geometry, we establish the explicit asymptotic behavior $λ_k \approx k^{\frac{2}{Q_0}}(\ln k)^{-\frac{2d_0}{Q_0}}$ as $k \to +\infty$, where $λ_k$ denotes the $k$-th Dirichlet eigenvalue of $\triangle_{X}$ on $Ω$, $Q_0$ is a positive rational number, and $d_0$ is a non-negative integer. Furthermore, we provide optimal bounds of index $Q_0$, which depend on the homogeneous dimension associated with the vector fields $X_1, X_2, \ldots, X_m$.
