Is there an example of a multivalued maximal monotone operator that is not the convex subdifferential of a proper convex lower semicontinuous? Besides, among these type of operators, are there any physically important? (describing any non-smooth dynamics of the real world). Thank you!



Any example of a multi-valued monotone maximal operator without subdifferential?
My prime example of such an operator comes from saddle point problems of the form
$$
\min\_x\max\_y F(x) + \langle Kx,y\rangle - G(y)
$$
with $F,G$ being two proper, convex, lower-semicontinuous functions defined on Hilbert spaces $X$ and $Y$, respectively, and $K:X\to Y$ linear and bounded. The Fenchel-Rockafellar optimality system is
$$
0\in \begin{pmatrix} \partial F(x) & -K^\*y\\ Kx & \partial G(y)\end{pmatrix}.
$$
The operator on the right hand side is indeed monotone as the sum of a subdifferential and a skew-symmetric one and it's also maximally so, since he both are and the second one is defined everywhere. It is not a subgradient of any function on $X\times Y$ since it's not symmetric.


