On Compact Riemannian Manifolds with Convex Boundary and Ricci Curvature Bounded from Below

We propose to study positive harmonic functions satisfying a nonlinear Neuman condition on a compact Riemannian manifold with nonnegative Ricci curvature and strictly convex boundary. A precise conjecture is formulated. We discuss its implications and present some partial results. Related questions are discussed for compact Riemannian manifolds with positive Ricci curvature and convex boundary.