On traces associated with the Dirichlet Laplacian (Q1095308)

The author proves the following theorem: Let \(\Delta_ D\) be the Dirichlet Laplacian for an open, bounded and connected set D in \(R^ m\), \(m=2,3,..\). with R-smooth boundary \(\partial D\) then for all \(t>0\) \[ (1)\quad | trace(e^{t\Delta_ D})-\frac{| D|}{(4\pi t)^{m/2}}+\frac{| \partial D|}{4.(4\pi t)^{(m-1)/2}}| \leq \frac{m^ 4}{\pi^{m/2}}\frac{| D| t^{1-m/2}}{R^ 2}, \] where \(| D|\) is the volume of D and \(| \partial D|\) is the area of \(\partial D.\) (A boundary \(\partial D\) of an open set D in \(R^ m\) is R-smooth if for each point \(x_ 0\in \partial D\) there exist two open balls \(B_ 1\), \(B_ 2\) with radii R such that \(B_ 1\subset D\), \(B_ 2\subset R^ m| \bar D\), \(\partial B_ 1\cap \partial B_ 2\ni x_ 0)\). By comparing the bound in (1) with the result of \textit{H. P. McKean} and \textit{I. M. Singer} [J. Differ. Geom. 1, 43-69 (1967; Zbl 0198.443)] it is shown that (1) is sharp. The proof relies on (i) the representation of the Dirichlet heat kernel in terms of conditional Wiener probabilities and (ii) decompositions of the brownian bridge. In addition theorems are proved for some unbounded domains. For example let \(D=\{(x,y):x>0,0<y<e^{-x}\}\) then for \(0<t<1\) \[ (2)\quad | \text{trace}(e^{t\Delta_ D})-\frac{1}{4\pi t}-\frac{1}{8.(\pi t)^{1/2}}\log t| \leq \frac{1}{t^{1/2}}. \]