Asymptotic property of eigenfunction of the Laplacian at the boundary (Q1324966)

Let \(M\) be a bounded region in \(\mathbb{R}^ n\) with smooth boundary, \(\lambda_ j\) the \(j\)-th eigenvalue and \(\{\varphi_ j\): \(j\in\mathbb{N}\}\) a complete orthogonal basis of eigenfunctions of \(-\Delta\) in \(M\) under Dirichlet condition. Then \[ \sum_{k=0}^ \infty e^{-\lambda_ j t} \partial_ \nu \varphi_ j(x)^ 2\sim \sum_{j=1}^ \infty C_ k(x) t^{{{k-n-3} \over 2}} \] holds when \(t\to 0\). The author also gives some conjectures and problems.