Heat asymptotics for Dirichlet elliptic operators with non-smooth coefficients

Article type: Research Article

Authors: Miyazaki, Yoichi

Affiliations: School of Dentistry, Nihon University, 1-8-13 Kanda-Surugadai, Chiyoda-ku, Tokyo 101-8310, Japan. Tel.: +81 3 3219 8166; Fax: +81 3 3219 8332; E-mail: [email protected]

Abstract: Let A be a 2mth-order elliptic operator in divergence form subject to the Dirichlet boundary conditions in a bounded Cρ domain Ω of Rn, whose coefficients are in Cσ 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.

Keywords: elliptic operator, Dirichlet boundary condition, heat kernel, asymptotic formula, trace, partition function, spectral function, counting function

DOI: 10.3233/ASY-2010-1024

Journal: Asymptotic Analysis, vol. 72, no. 3-4, pp. 125-167, 2011