加菲尔德不等式与在无界领域中德拉姆复形的闭范围性质
The Closed Range Property and Gaffney's Inequality of the De Rham Complex in Unbounded Domains
AI总结：
本文研究了在无界领域中，通过加菲尔德不等式和生物利普希茨变换稳定性，得出旋转算子在两个方向受限时的闭范围性质，并应用于麦克斯韦算子的谱隙分析及稳定性研究。
摘要：
经典的庞加莱估计确立了在无权$L^2(Ω)$-空间中，只要$Ω⊆\mathbb{R}^3$位于一个薄片内，即$Ω$在一个方向上有界，梯度的范围是闭合的。本文的主要观察是，在$Ω$在两个方向上有界时，提供旋转算子的闭范围结果。在此过程中，我们通过描述基础域的有界方向，对所有原始和对偶德拉姆复形的微分算子的闭范围结果进行了特征化。作为主要应用，我们得到了麦克斯韦算子在0附近存在谱隙，从而允许在电导率充分阻尼的情况下，麦克斯韦方程组的解具有指数稳定性。我们的结果基于加菲尔德不等式和其在无界（简单）域中的有效性，以及闭范围结果在生物利普希茨正则变换下的稳定性。后者技术在附录中已详细说明；对于加菲尔德估计的结果，我们使用标准结果提供简单证明。此外，我们还将展示非平凡的例子和一个仅在一个方向上有界的集合上具有混合边界条件的旋转算子的闭范围结果。
英文摘要：
The classical Poincaré estimate establishes closedness of the range of the gradient in unweighted $L^2(Ω)$-spaces as long as $Ω\subseteq\mathbb{R}^3$ is contained in a slab, that is, $Ω$ is bounded in one direction. Here, as a main observation, we provide closed range results for the $\operatorname{rot}$-operator, if (and only if) $Ω$ is bounded in two directions. Along the way, we characterise closed range results for all the differential operators of the primal and dual de Rham complex in terms of directions of boundedness of the underlying domain. As a main application, one obtains the existence of a spectral gap near the $0$ of the Maxwell operator allowing for exponential stability results for solutions of Maxwell's equations with sufficient damping in the conductivity. Our results are based on the validity of Gaffney's (in)equality and the transition of the same to unbounded (simple) domains as well as on the stability of closed range results under bi-Lipschitz regular transformations. The latter technique is well-known and detailed in the appendix; for the results concerning Gaffney's estimate, we shall provide accessible, simple proofs using mere standard results. Moreover, we shall present non-trivial examples and a closed range result for $\operatorname{rot}$ with mixed boundary conditions on a set bounded in one direction only.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
