Given a first order elliptic operator $D:\Gamma(X; E)\to \Gamma(X; F)$ where $X$ is a closed manifold, and $E\to X, F\to X$ vector bundles, we know that $D$ induces a Fredholm operator
between the spaces of Sobolev sections $D: W^{k+1,2}(X;E)\to W^{k,2}(X;F) $.
Therefore we can compute its index which is a topological quantity given by the celebrated Atiyah-Singer index theorem.
Now suppose that $X$ is a compact manifold with boundary, and consider $D$ as an operator
$$D: W_0^{k+1,2}(X;E)\to W_0^{k,2}(X;F) $$
where $W_0^{k,2}(X;E)$ is the space of Sobolev sections vanishing over $\partial X$: the completion of $C^\infty_0(\mathrm {int} (X); E)$ in the Sobolev norm $\| \cdot \|_{W^{k,2}} $.
$D$ is again a Fredholm operator.
Is there a formula for its index?
Note: this is different from the case considered by Atiyah, Patodi and Singer; in my case I impose a boundary condition $f|_{\partial X} =0$, in their case they impose $\Pi_{-} f|_{\partial X} = 0$ where $\Pi_{-}$ is the $L^2$-projector to the negative eigenspace of the boundary operator $L$ (i.e. over a collar of the boundary $D = \sigma(\frac {\partial}{dt} + L$)).
Best Answer
Local boundary conditions such as the Dirichlet condition you mention were considered in Atiyah-Bott, The index problem for manifolds with boundary. 1964 Differential Analysis, Bombay Colloq., 1964 pp. 175–186 Oxford Univ. Press, London. (There is also a chapter by Atiyah in Palais's Seminar on the Atiyah-Singer Index Theorem with similar material.)
My understanding is that such boundary conditions do not typically lead to elliptic problems, eg for the signature operator. This was one of the motivations for the Atiyah-Patodi-Singer approach.