Asymptotic expansion of the trace of the heat kernel associated to the Dirichlet-to-Neumann operator

作者：Liu Genqian^*

来源：Journal of Differential Equations, 2015, 259(7): 2499-2545.

DOI：10.1016/j.jde.2015.03.029

• Dirichlet-to-Neumann operator
• Steklov eigenvalue
• Heat kernel associated to Dirichlet-to-Neumann operator
• Asymptotic expansion
• Curvature

For a given bounded domain Omega with smooth boundary in a smooth Riemannian manifold (M, g), by decomposing the Dirichlet-to-Neumann operator into a sum of the square root of the Laplacian and a pseudodifferential operator, and by applying Grubb's method of symbolic calculus for the corresponding pseudodifferential heat kernel operators, we establish a procedure to calculate all the coefficients of the asymptotic expansion of the trace of the heat kernel associated to Dirichlet-to-Neumann operator as t --> 0(+). In particular, we explicitly give the first four coefficients of this asymptotic expansion. These coefficients provide precise information regarding the area and curvatures of the boundary of the domain in terms of the spectrum of the Steklov problem.

• 出版日期2015-10-5
• 单位北京理工大学