ON THE PRESCRIBED MEAN CURVATURE PROBLEM ON THE STANDARD n-DIMENSIONAL BALL

Abstract

In this paper, we consider the problem of existence of confor- mal metrics with prescribed mean curvature on the unit ball of Rn, n � 3. Under the assumption that the order of flatness at critical points of pre- scribed mean curvature function H(x) is � 2)1,n 2), we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula. 1. Introduction and main result In this article, we consider the problem of existence of conformal scalar flat metric with prescribed boundary mean curvature on the standard n-dim- ensional ball. Let B n be the unit ball in R n , n ≥ 3, with Euclidean metric g0. Its boundary will be denoted by S n−1 and will be endowed with the standard metric still denoted by g0. Let H : S n−1 → R be a given function, we study the problem of finding a conformal metric g = u 4 n−2g 0 such that Rg = 0 in B n and hg = H on S n−1 . Here Rg is the scalar curvature of the metric g in B n and hg is the mean curvature of g on S n−1 . This problem is equivalent to solving the following nonlinear boundary value equation: (P) ( �u = 0 in B n ∂u ∂ν + n − 2 2 u = n − 2 2 Hu n n−2 on S n−1 ,