If X is a smooth manifold with boundary with dimension m, and P is an elliptic partial differential operator on X with smooth coefficients, and f is a locally integrable function on X with Pf=0 in the sense of distribution theory on the interiour of X, is it then true that f is smooth up to the boundary (even without any boundary conditions)?

I think my confusion arises from the fact that the charts on X, which are open subsets of half plane, need not have a smooth boundary (as subsets of R^m).

If X is a smooth manifold with boundary and of dimension m, and P is an elliptic partial differential operator on X with smooth coefficients, and f is a locally integrable function on X with Pf=0 in the sense of distribution theory on the interior of X, is it then true that f is smooth up to the boundary (even without any boundary conditions)?

If X is the closure of an open bounded subset of R^m, then I believe the result is correct. So my confusion arises from the fact that the charts on X, which are open subsets of half plane rather then R^m, need not have a smooth boundary (as subsets of R^m).