Title:

On the prescribed mean curvature problem on the standard ndimensional ball

In this thesis, we consider the problem of existence of conformal scalar flat metric with prescribed boundary mean curvature on the standard ndimensional ball. Let B[superscript n] be the unit ball in R[superscript n], n ≥ 3, with Euclidean metric g[subscript 0]. Its boundary will be denoted by S[superscript n1] and will be endowed with the standard metric still denoted by g[subscript 0]. Let H : S[superscript n1] → R be a given function, we study the problem of finding a conformal metric g = u 4/n2 g[subscript 0] such that R[subscript g] = 0 in B[superscript n] and h[subscript g] = H on S[superscript n1]. Here R[subscript g] is the scalar curvature of the metric g in B[superscript n] and h[subscript g] is the mean curvature of g on S[superscript n1]. This problem is equivalent to solving the following nonlinear boundary value equation: (see PDF for equation) where v is the outward unit vector with respect to the metric g[subscript 0]. In general there are several difficulties in facing this problem by means of variational methods. Indeed, in virtue of the noncompactness of the embedding H[superscript 1](B[superscript n]) → L 2(n1)/n2 (∂B[superscript n]), the EulerLagrange functional J associated to the problem, does not satisfy the PalaisSmale condition, and that leads to the failure of the standard critical point theory. One part of this thesis deals with the case where H is a Morse function satisfying a non degeneracy condition. Using an algebraic topological method and the tools of the theory of the critical points at infinity, we provide a variety of classes of functions that can be realized as the mean curvature on the boundary of the the ndimensional balls. The other part deals with the case where the non degeneracy condition is not satisfied and replaced by the so called βflatness condition. In this case, we give precise estimates on the losses of the compactness and we identify the critical points at infinity of the variational problem. Then, we establish under generic boundary condition a Morse inequalities at infinity, which give a lower bound on the number of solutions to the above problem.
