欧几里得空间R^n中有界域的庞加莱常数估计

中文摘要 AI 辅助

我们研究空间R^n中（有限维n>1）有界、单连通、凸且C2光滑的域。对于此类域，我们利用直径来寻找包含该域的球，并选择直径最小的球。通过一个简单的变换，我们能够将我们的域映射到开单位球内的一个域上。我们推导了这些域上标量和向量函数的庞加莱常数估计。主要成分分别是具有消失Dirichlet迹的Laplacian或Stokes算子的第一特征值（我们使用文献[21]的结果）。此外，我们应用了（标量）Laplace和Stokes算子的第一特征值与一个特征函数之间的关系，参见[21]。我们对庞加莱常数的估计是新的。空间的维数n对我们估计的质量有显著影响。这通过一个例子加以说明。

英文摘要

We study bounded, simply connected, convex, and C2-smooth domains in the space R^n for finite n greater than 1. For such domains we use the diameter to find balls with the property that they are included in this ball, while the diameter of the ball is chosen to be minimal. With a simple transformation we are able to map our domain onto a domain inside the open unit ball. We derive estimates for the Poincare constants of scalar and vector functions on those domains. The main ingredient is the first eigenvalue of the Laplacian or the Stokes operator with vanishing Dirichlet traces, respectively (we use results of [21]). Additionally we apply the relation of the first eigenvalues and one eigenfunction of the (scalar) Laplace and the Stokes operator cf. [21]. Our estimates for the Poincare constants are new. The dimension n of the space has significant effect on the quality of our estimates. This is illustrated with the help of an example.

发表机构

• Otto-von-Guericke-Universität Magdeburg（马格德堡奥托·冯·格里克大学）
• KIT（卡尔斯鲁厄理工学院）

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