Gilkey-Smith边值问题的热核渐近性
Heat Kernel Asymptotics of the Gilkey-Smith Boundary Value Problem
摘要：
紧黎曼流形上的规范理论在边界上的表述导致了具有Gilkey-Smith边界条件的偏微分算子，其特殊性质是在边界上同时出现法向和切向导数。与标准的Dirichlet或Neumann边界条件不同，这个边值问题不是自动椭圆的，而是在边界算子的某些条件下变成椭圆的。本文研究了Laplace型算子的Gilkey-Smith边值问题，得到了椭圆性的一个简单判据。计算了热核迹的渐近展开式的第一非平凡系数，得到了热核对角线的局部领先渐近性。结果表明，在非椭圆情形下，热核对角线在边界附近不可积，这反映了热核不是迹类的事实。我们将这种分析应用到一般的线性玻色子规范理论中，并找到了一个椭圆性的显式条件。
英文摘要：
The formulation of gauge theories on compact Riemannian manifolds with boundary leads to partial differential operators with Gilkey--Smith boundary conditions, whose peculiar property is the occurrence of both normal and tangential derivatives on the boundary. Unlike the standard Dirichlet or Neumann boundary conditions, this boundary-value problem is not automatically elliptic but becomes elliptic under certain conditions on the boundary operator. We study the Gilkey--Smith boundary-value problem for Laplace-type operators and find a simple criterion of ellipticity. The first non-trivial coefficient of the asymptotic expansion of the trace of the heat kernel is computed and the local leading asymptotics of the heat-kernel diagonal is also obtained. It is shown that, in the non-elliptic case, the heat-kernel diagonal is non-integrable near the boundary, which reflects the fact that the heat kernel is not of trace class. We apply this analysis to general linear bosonic gauge theories and find an explicit condition of ellipticity.
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