Let (M, g) be a compact Riemannian manifold with boundary \(\partial M\) . Given a function f on \(\partial M\) , we consider the problem of finding a conformal metric of g with zero scalar curvature in M and prescribed mean curvature f on \(\partial M\) . Through the construction of local test functions, we resolve most of the remaining open cases from Escobar’s work [10] and establish new solvability conditions.
