# Existence and uniqueness of the Robin problem on a compact, smooth Riemannian manifold with boundary

Let $(\overline{M},g)$ a compact, smooth, Riemannian manifold with boundary $\partial M \in C^\infty$. By $\nu_g$ we denote the normalvectorfield, by $\nabla_g$ the gradient and by $\Delta_g$ the Laplace-Beltrami operator induced by $g$. Has the problem \begin{align} \begin{cases} \Delta_g u = 0, \quad in \ M ,\\ g(\nabla_g u , \nu_g ) + \lambda u = \varphi, \quad on \ \partial M \end{cases} \end{align} a unique solution for $\varphi \in C(\partial M)$ and $\lambda > 0$?