This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain $U \subset \mathbb{R}^N$.
1) In principle the eigenvalues of such operator are defined as those $\lambda \in \mathbb{C}$ for which \begin{equation} N(\Delta-\lambda I) \neq \lbrace 0 \rbrace \end{equation} where $N$ denotes the nullspace.
2) Now many books, e.g. [1] say that the eigenvalues of the Laplacian are those $\lambda \in \mathbb{C}$ for which the Dirichlet problem on U\begin{equation} \begin{cases} \Delta u = \lambda u & \text{in } U \\ u = 0 & \text{on } \partial U\end{cases} \end{equation} has non-trivial solutions.
With the first definition any complex value $\lambda$ would be an eigenvalue of the Laplacian, because $e^{\sqrt{\lambda} x} \in N(\Delta - \lambda I)$.
With the second definition the eigenvalues would form a discrete set of real, negative eigenvalues according to [1, Chap. 6.5, Thm.1].
My Question is twofold:
1) How do define eigenvalues of the Laplacian? If it is the second way, why do you (of all things) impose zero boundary conditions? Wouldn't you actually have to call them "eigenvalues of the Dirichlet problem" then?
2) How do the two definitions fit together? Do you implicitly view $\Delta$ as an operator $C^2_0(U) \to C(U)$ in the second case (where the zero subscript denotes zero boundary conditions)?
[1] Evans, L. C.: Partial differential equations, 19 in "Graduate Studies in Mathematics". American Mathematical Society, Providence, RI, 2nd edition, 2010.
Both define eigenvalues (and eigenvectors), however for different spaces: in the first case we talk about the space of $C^2$ functions on $U$, in the second case we talk about the space of $C^2$ functions on $U$ that vanish on the boundary.