The Boundary Behavior of Heat Kernels of Dirichlet Laplacians

AbstractWe 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), ρ(x)ρ(y)t∧1c1tn/2e−c2∣x−y∣2/t⩽G(x,t;y,0)⩽ρ(x)ρ(y)t∧11c1tn/2e−∣x−y∣2/(c2t) 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