# A quantity associated to a compact Riemannian manifold with boundary(The pair of Laplacian)

Let $$M$$ be a Riemannian manifold with boundary $$\partial M$$. Are the following operators, Fredholm operators?Is there a geometric terminology and geometric interpretation for the fredholm index of these operators?

1) The first operator is the pair of Laplacian $$(\Delta,\Delta_{\partial})$$ defined on $$M$$ and $$\partial M$$, where $$\Delta_{\partial}$$ is the Laplacian associated to the Riemannian metric which $$\partial M$$ inherits from $$M$$.

2)The second operator is the standard Cauchy problem for Harmonic functions, $$\Delta u=0$$ but the boundary condition is $$\Delta_{\partial} u$$.