Properties of heat kernels

Abstract

We prove two kinds of results related to the asymptotic behavior of the Dirichlet or Neumann heat kernels. The first concerns the monotonicity of the Neumann heat kernel toward the boundary. We show that the diagonal of the radial Neumann heat kernel of the ball is increasing toward the boundary if and only if the dimension of the space is bigger than two. The second problem is to give good bounds for the first few eigenvalues of the Dirichlet Laplacian on bounded sets. This gives long time asymptotics of the Dirichlet heat kernel. We prove new upper and lower bounds for triangular domains involving various geometric measurements of the domain