Abstract :
We establish the global boundary behavior of Dirichlet heat kernels on some unbounded domains. These, combined with the interior estimate in the recent paper by Grigoryan and Saloff-Coste (Comm. Pure Appl. Math 55(1) (2002) 93), provide a complete and qualitatively sharp description of heat kernels G of Dirichlet Laplacians on some unbounded C1,1 domains D⊂M. Here M is certain complete noncompact manifold with nonnegative Ricci curvature. For example, there exist positive constants c1,c2 depending on the unbounded domain D, M such that, for ρ(x)=dist(x,∂D), ρ(x)t∧1 ∧1ρ(y)t∧1 ∧1c1e−c2d(x,y)2t|B(x,t)|⩽G(x,t;y,0)⩽ρ(x)t∧1 ∧1ρ(y)t∧1 ∧1e−d(x,y)2c2tc1|B(x,t)|for all x,y∈D and t>0. Existence of these global bounds had been an unresolved problem even in the Euclidean case.
