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 $\Delta_{\partial} u$.


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