Does de Rham theorem hold for manifolds with boundary? I am following the J.Lee's book "Introduction to Smooth Manifolds", 2nd ed., page 480-486 to learn the de Rham theorem. It is proven on manifolds without boundary, which makes me curious about whether it holds for manifolds with boundary. At least in the part proving de Rham cohomology is isomorphic to the $C^\infty$ singular cohomology, I did not see a step at first glance where the manifold in question must have no boundary. ("At first glance" means I have checked it did not cite theorems like Whitney approximation theorem that specifically requires the target to be without boundary, but I did not check if all the computation/formula really works for manifolds with boundary.) I also try to search for more sources, and what I have found only talk about the de Rham theorem in no-boundary case. 
Thus I wonder if de Rham theorem holds for manifolds with boundary. If it fails, is it still true that de Rham cohomology is isomorphic to the $C^\infty$ singular cohomology?
 A: Yes, it holds for manifolds with boundary. One way to see this is to note that if $M$ is a smooth manifold with boundary, then the inclusion map  $\iota\colon \text{Int}\  M\hookrightarrow M$ is a smooth homotopy equivalence (Thm. 9.26 in ISM), which therefore induces isomorphisms on de Rham cohomology (Thm. 17.11).  Since $\iota$ also induces isomorphisms on singular cohomology, and the de Rham homomorphism (integration over chains) commutes with inclusion, the result follows. 
(It would have been a good idea to include this in the book, either as a corollary or as a problem. I don't know why I didn't.)
A: Ray and Singer prove a version of de Rham's Theorem for manifolds with boundary (Proposition 4.2 of R-torsion and the Laplacian on Riemannian manifolds, adv. in math. 1971). They need boundary conditions to establish the Theorem. Actually the previous answer looks incorrect. The result obtained by Ray and Singer is:          Let W be an h-cobordism whose boundary is M_1 U M_2. Consider relative boundary conditions on M_1 and absolute boundary conditions on M_2. Denote by H the space of harmonic forms on W. Then H is isomorphic to H^q(W,M_1) (relative cohomology). You may also want to check expression (2.7) of W. Muller, on page 735 of "Analytic Torsion and R-torsion for unimodular representations", J. Amer. Math. Soc. 1993. Muller's statement does not require the manifold to be an h-cobordism. But I am not an expert, I was only searching for an alternative proof for the Theorem, perhaps in a book, since I was having difficulty to understand these references.
