Boundary Value Problems for some Fully Nonlinear Elliptic Equations

Let (M, g) be a compact Riemannian manifold of dimension n ≥ 3 with boundary ∂M . We denote the Ricci curvature, scalar curvature, mean curvature, and the second fundamental form by Ric, R , h, and Lαβ , respectively. The Yamabe problem for manifolds with boundary is to find a conformal metric ĝ = eg such that the scalar curvature is constant and the mean curvature is zero. The boundary is called umbilic if the second fundamental form Lαβ = μggαβ. For example, a totally geodesic boundary is umbilic with zero principal curvatures. In [8], it was proved by Escobar that for locally conformally flat compact manifolds with umbilic boundary (and some other cases), the Yamabe problem is solvable. As for the nonlinear version of the Yamabe problem, we consider the Schouten tensor defined as Ag = 1 n− 2 (Ric− R 2(n− 1) g).