Xiaoping Yang

ON STATIONARY HYPERSURFACES IN EUCLIDEAN SPACES

Abstract Some techniques in the Geometric Measure Theory are used to study the hypersurfaces in Euclidean spaces and some fundamental properties with this subject are discussed in this article.

DECOMPOSITION OF BV FUNCTIONS IN CARNOT-CARATHÉODORY SPACES

Abstract The aim of this paper is to get the decomposition of distributional derivatives of functions with bounded variation in the framework of Carnot-Caratheodory spaces (C-C spaces in brievity) in which the vector fields are of Carnot type. For this purpose the approximate continuity of BV functions is discussed first, then approximate differentials of L1 functions are defined in the case that vector fields are of Carnot type and finally the decomposition Xu = ∇u · Ln + Xsи is proved, where и ∈ BVx(Ω) and ∇u denotes the approximate differential of u.

WEAK SOLUTIONS OF MONGE-AMPÈRE TYPE EQUATIONS IN OPTIMAL TRANSPORTATION

Abstract This paper concerns the weak solutions of some Monge-Ampere type equations in the optimal transportation theory. The relationship between the Aleksandrov solutions and the viscosity solutions of the Monge-Ampere type equations is discussed. A uniform estimate for solution of the Dirichlet problem with homogeneous boundary value is obtained.

EXISTENCE OF MINIMISERS FOR A CLASS OF FREE DISCONTINUITY PROBLEMS IN THE HEISENBERG GROUP H n

Abstract The purpose of this paper is to prove existence of minimisers of the functionaln J ( K , u ) : = ∫ f ( L u ) d x + α ∫ ‖ u − g ‖ q d x + β D d Q − 1 ( K ∩ Ω ) , where Ω is an open set of the Heisenberg group Hn, K runs over all closed sets of Hn, u varies inn C H 1 ( Ω /K), α , β >0, q ≥ 1 , g ∈ L q ( Ω ) ∩ L ∞ ( Ω ) and f : R2n → R is a convex function satisfying some structure conditions (H1)(H2)(H3) (see below).