Maximal monotone operator without convex domain?

Question

I'm looking for an example of a (multi-valued) maximal monotone operator $A$ mapping a Banach space $X$ into its dual $X^*$ such that the domain $D(A)=\{x\in X: Ax\neq\emptyset\}$ is not convex. Preferably, the example should be simple (maybe with $X=\mathbb{R}^2$).

Thanks a lot in advance for suggestions

Answer 1

The convex subdifferential of a proper convex lower semicontinuous function defined in a Banach space is a maximal monotone operator (that is a result of Rockafellar). There are several examples of subdifferentials without a convex domain, the one written below being taken from Rockafellar's book "Convex Analysis" on page 218.

Let $f(\xi_1,\xi_2)=\max \{g(\xi_1),|\xi_2|\}$, where $g(\xi_1)=1-\xi_1^{1/2}$, if $\xi_1\ge 0$, $g(\xi_1)=+\infty$, if $\xi_1< 0$.

The domain of $f$ is the right closed half-plane, and f is subdifferentiable everywhere on this half-plane except in the relative interior of the linesegment joining $(0,1)$ and $(0,-1)$.

You can build your own example. Take into consideration that for $\mathbb {R}^n$ all maximal monotone operators have the closure of their domains convex.