Using the divergence theorem or Green’s formula, we obtain that the
- Euclidean Laplacian,
- Dirichlet Laplacian on bounded domains or compact manifolds,
- Neumann Laplacian on bounded domains or compact manifolds,
- Laplacian on compact manifolds without boundary,
suffices the following properties:
- self-adjoint, and thus the eigenspaces (if any exists) are orthogonal to each other,
- the spectrum is non-negative,