Basic question about the invertibility of a function $F: \mathbb{R}^2 \to \mathbb{R}^2$

Question

Let $\phi :\mathbb{R}^2 \to \mathbb{R}^2$ be a convex function and let's define a new map $F :\mathbb{R}^2 \to \mathbb{R}^2$ as follows: $$ F(x, y) = (x, y) + \nabla \phi (x, y). $$ How can I prove that it's a diffeomorphism? Using convexity, I can easily see that it is injective and it has invertible Jacobian. But I don't know how to see the surjectivity.

Answer 1

Minty's theorem. Check out Chapter 21 of Convex Analysis and Monotone Operator Theory in Hilbert Spaces (Bauschke and Combettes). The subdifferential of a convex function is maximally monotone (20.40). By Minty's theorem, this means $\text{ran}(\partial f+I)=\mathcal{H}$.