论文标题
传输特征值的Weyl定律和广义传播特征功能的完整性
The Weyl law of transmission eigenvalues and the completeness of generalized transmission eigenfunctions
论文作者
论文摘要
传输问题是一个两个二阶椭圆方程的系统，该系统的两个未知数配备了cauchy数据。经过四十年的散射理论的研究，现在已知该问题的光谱特性取决于边界附近系数之间的对比度。以前，我们确定了与Agmon，Douglis和Nirenberg引起的著名补充条件有关的大量各向异性系数的特征值的离散性。在这项工作中，我们为本类系数的特征值建立了特征值的Weyl定律，并在对系数的连续性的额外假设下为这类系数建立了本类系数的特征函数的完整性。该分析是新的，并基于此处建立的传输问题的$ l^p $规则理论。它还涉及光谱理论对希尔伯特·施密特（Hilbert Schmidt）操作员的微妙应用。我们的工作在文献中扩展了很大的已知结果，这些结果主要专门用于$ c^\ infty $ cefficients的各向同性案例。
The transmission problem is a system of two second-order elliptic equations of two unknowns equipped with the Cauchy data on the boundary. After four decades of research motivated by scattering theory, the spectral properties of this problem are now known to depend on a type of contrast between coefficients near the boundary. Previously, we established the discreteness of eigenvalues for a large class of anisotropic coefficients which is related to the celebrated complementing conditions due to Agmon, Douglis, and Nirenberg. In this work, we establish the Weyl law for the eigenvalues and the completeness of the generalized eigenfunctions for this class of coefficients under an additional mild assumption on the continuity of the coefficients. The analysis is new and based on the $L^p$ regularity theory for the transmission problem established here. It also involves a subtle application of the spectral theory for the Hilbert Schmidt operators. Our work extends largely known results in the literature which are mainly devoted to the isotropic case with $C^\infty$-coefficients.
