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 $\lambda > 0$?