Concavity of Cobb-Douglass Utility Function on Non-Open set
I assume the notation $\mathbb R^2_+$ refers to $[0,\infty)^2$.

Note that the set on which a function is defined need not be the same set on which a function is differentiable. In particular, it's typical that differentiability requirements are imposed on open sets (see e.g. the fundamental theorem of calculus). This is because defining differentiability on boundary points requires extra care that's usually unnecessary for the issue at hand (such as defining concavity of a function via the Hessian matrix). 

The Cobb-Douglas function is defined on $\mathbb R^2_+=[0,\infty)^2$, (continuous also on that domain), and differentiable on $(0,\infty)^2$, which is open and convex.