The boundary behavior of heat kernels of Dirichlet Laplacians

Abstract

We prove the following complete and qualitatively sharp description of heat kernels G of Dirichlet Laplacians on bounded C1,1 domains D. There exist positive constants c1, c2 and T > 0 depending on D such that, for ρ(x) = dist(x, ∂D), for all x, y ∈ D and 0 < t ≤ T. The upper bound is well known since the 1980s (E.B. Davies, J. Funct. Anal. 71 (1987), 88-103) however, the existence of the lower bound had been an open question since then. (Bounds when t > T are known.) Bounds when D is unbounded are also given. © 2002 Elsevier Science (USA).