Variational Formulas And Basic Estimates Of The Optimal Constant In Hardy-type Inequalities On Trees

The Hardy-type inequality describes that the Lq(?)-norm of an absolutely continuous function f can be controlled from above by the Lp(v)-norm of its derivative f' and a con-stant, it is a basic tools in the field of Probability Theory, Functional Analysis, Harmonic Analysis and PDE. In the paper, we concentrated on estimating the optimal constant in general Hardy-type inequalities on trees.We divide the Hardy-type inequality into 4 cases according to the boundary condi-tions, which are DN-, ND-, DD- and NN-case boundary condition. Here, "D" is Dirichlet boundary (absorbing boundary),"N" is Neumann boundary (reflection boundary). In this paper, we focus on the optimal constant of Hardy-type inequalities on trees with DN- and NN-case boundary conditions.In chapter one, we will introduce the background on the subject.In chapter two, first, we study the general Hardy-type inequalities on trees with DN-case boundary conditions, the variational formulas and the basic estimation of the optimal constant are introduced, as well as the approximation procedure. The main re-search method benefited from the works of Prof. Chen (2004), especially the work on the estimation of convergence rate of symmetric Markov process. Part results of this paper can be considered as the further discussion of the results on the convergence rate of the birth and death process and the estimates on the first Dirichlet eigenvalue of p-Laplacian operators in Zhang (2013) and Wang (2015). Second, we will present Lipschiz norm of the p-Laplacian (p ? 2) operator and discuss the relationship between the estimates on the first Dirichlet eigenvalue of p-Laplacian (p ? 2) operator (i.e. Hardy-type inequalities optimal constant provided p = q) and Lipschiz norm, ? norm of p-Laplacian (p ? 2)operator by constructing some proper functional space on trees.In chapter three, we first study the optimal constants of the Hardy-type inequality with NN- and NN-case boundary conditions, the relationship of the two optimal constants.To optimize the relationship, we describe their by using the median on trees. Second, we will present Lipschiz norm of the p-Laplacian (p ? 2) operator and discuss the relationship between the estimates on the fist Dirichlet eigenvalue of p-Laplacian (p?2) operator and Lipschiz norm of p-Laplacian (p ? 2) operator on trees.In chapter four, we will present a short summary and list some problems that need to be solved for further study.