The Nirenberg Problem in a Domain with Boundary

There has been much work on the Nirenberg problem: which function K(x) on S is the scalar curvature of a metric g on S pointwise conformal to the standard metric g0? It is quite natural to ask the following question on the half sphere S−: which function K(x) on S− is the scalar curvature of a metric g on S− which is pointwise conformal to the standard metric g0 with ∂S− being minimal with respect to g? For n = 2, this has been studied by J. Q. Liu and P. L. Li in [LL]. In this note we study the higher dimensional cases along the lines of [L1-2]. For much work on the Nirenberg problem see, for example, [L1-2] and the references therein. See also some more recent work in [CL1], [HL], [Bi1-2], [SZ], [B], [ChL] and [CL2]. For n ≥ 3, by writing g = ug0, the problem is equivalent to solving the following Neumann problem on S− = {(x1, . . . , xn+1) ∈ S | xn+1 < 0}: