I am asking about the surjectivity of the Robin-boundary trace operator i.e., \begin{align} \mathcal{L}:& C^2(\overline{\Omega}) \to C(\partial \Omega)\\ \mathcal{L} \varphi:&=\partial_n \varphi+ \beta \varphi, \end{align}
where $C(\partial \Omega)$ (resp. $C^2(\overline{\Omega})$) is the space of continuous (resp. classe $C^2$) functions on (the sufficiently smooth boundary $\partial \Omega$ of the bounded domain $\Omega$, $\beta \geq 0$ and $\partial_n$ is the normal derivative over a normal vector n. My question is whether this operator is surjective or not? In particular it is also surjective if we take the case of Neumann boundary conditions (i.e., with $\beta=0$)? If someone have a reference or any idea/a counter-example?
Remark We can replace $C^2(\overline{\Omega})$ by the domain of the Laplace operator (without boundary conditions) in the space $C(\overline{\Omega})$