Conformal Scalar-Flat Metrics with Prescribed Boundary Mean Curvature

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