Heat kernel estimates on manifolds with ends with mixed boundary condition

Abstract

We obtain two-sided heat kernel estimates for Riemannian manifolds with ends with mixed boundary condition, provided that the heat kernels for the ends are well understood. These results extend previous results of Grigor'yan and Saloff-Coste by allowing for Dirichlet boundary condition. The proof requires the construction of a global harmonic function which is then used in the h-transform technique. introductionIn [6], Alexander Grigor'yan and the second author initiated the study of twosided heat kernel estimates on weighted complete Riemannian manifolds with finitely many nice ends,The components M i of this connected sum are, themselves, assumed to be weighted complete Riemannian manifolds. The main assumption is that, on each M i , the heat kernel p Mi (t, x, y), is well understood in the sense that it satisfies a classical-looking two-sided Gaussian estimate, uniformly at all times and locations. Equivalently ([4, 17, 18]), the volume functions of these manifolds, M i , 1 ≤ i ≤ k, are uniformly doubling at all scales and locations AND their geodesic balls satisfy a Neumann-type Poincaré inequality, uniformly at all all scales and locations. These are very strong hypotheses, and, in certain cases, additional more technical hypotheses are needed. The results of [6] are sharp twosided estimates on the heat kernel of M. The most basic case illustrating these results is when M i = R N for some N, and, more generally, M i = R ni × S N −ni , for some N and n i , 1 ≤ n i ≤ N . These basic cases were new and already plenty challenging at the time [6] was published. They are richer than they appear if one takes into consideration the variation afforded by the weight functions. In addition, the results hold without change when the term "complete Riemannian manifold" is interpreted in the context of manifolds with boundary. Complete, then, means metrically complete, and the heat equations and heat kernels on M and on the M i , 1 ≤ i ≤ k, are all taken with Neumann boundary condition. So, for instance, the results of [6, 10] cover the solid three-dimensional body in Figure 1. (This figure created by A. Grigor'yan appears in [10].)The aim of the present work is to initiate the study of the case when the heat equation on the complete manifold M (with boundary) above is taken with mixed boundary condition: Neumann on some part of the boundary and Dirichlet on the rest of the boundary.