Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces

In this paper we present a survey of the joint program with Fabrice Baudoin originated with the paper \cite{BG1}, and continued with the works \cite{BG2}, \cite{BBG}, \cite{BG3} and \cite{BBGM}, joint with Baudoin, Michel Bonnefont and Isidro Munive.Comment: arXiv admin note: substantial text overlap with arXiv:1101.359

The mixed problem for the Laplacian in Lipschitz domains

We consider the mixed boundary value problem or Zaremba's problem for the Laplacian in a bounded Lipschitz domain in R^n. We specify Dirichlet data on part of the boundary and Neumann data on the remainder of the boundary. We assume that the boundary between the sets where we specify Dirichlet and Neumann data is a Lipschitz surface. We require that the Neumann data is in L^p and the Dirichlet data is in the Sobolev space of functions having one derivative in L^p for some p near 1. Under these conditions, there is a unique solution to the mixed problem with the non-tangential maximal function of the gradient of the solution in L^p of the boundary. We also obtain results with data from Hardy spaces when p=1.Comment: Version 5 includes a correction to one step of the main proof. Since the paper appeared long ago, this submission includes the complete paper, followed by a short section that gives the correction to one step in the proo

A probabilistic proof of the boundary Harnack principle

The main purpose of this paper is to give a probabilistic proof of Theorem 1.1, one using elementary properties of Brownian motion. We also obtain the fact that the Martin boundary equals the Euclidean boundary as an easy corollary of Theorem 1.1. The boundary Harnack principle may be viewed as a Harnack inequality for conditioned Brownian motion; as an application we prove some new probability bounds for conditioned Brownian motion in Lipschitz domains.Research partially supported by NSF grants DMS 8701073 and DMS 8901255

Multiple non-negative solutions to a semilinear equation on Heisenberg group with indefinite nonlinearity

This paper is concerned with the existence and multiplicity of non-negative solutions to the semilinear equation (Formula presented.) in a bounded domain Ω⊂HN with Dirichlet boundary conditions. Here HN is the Heisenberg group and 2♯=2q/(q-2) is the critical exponent of the Sobolev embedding on the Heisenberg group. The function K(ξ) may be sign changing on Ω. Using the variational method, we prove that this problem has at least two non-negative solutions provided μ, α, and K(ξ) satisfy some conditions.publishe