Heat asymptotics for Dirichlet elliptic operators with non-smooth coefficients

Let A be a 2mth-order elliptic operator in divergence form subject to the Dirichlet boundary conditions in a bounded C^ρ domain Ω of Rⁿ, whose coefficients are in C^σ with given σ>0, where ρ=1 if 0<σ≤1 and ρ=σ+1 if σ>1. Then we investigate the asymptotic formula for the partition function, namely the trace of the kernel of exp (−tA) as t→+0, and show that the well-known asymptotic formula for the C^∞ case remains valid up to the order according to σ. We also give the asymptotic formula for the heat kernel in Rⁿ, Rⁿ₊ and a bounded C¹ domain.