The following is a classical definition due to Morrey: Let $\Omega \subset \mathbb{R}^n$ be a nice enough, bounded domain and $f: \mathbb{R}^{m \times n} \to \mathbb{R}$ with some reasonable growth conditions. We call $f$ quasiconvex if $$ \inf_{\varphi \in C_0^\infty(\Omega)} \int_\Omega f(A+D\varphi) dx = \int_\Omega f(A) dx $$ for all $A\in \mathbb{R}^{m\times n}$. Or equivalently if $$\int_{\mathbb{R}^{m\times n}} f(\lambda) d\nu(\lambda) \geq f(\bar{\nu})$$ for all homogeneous $W^{1,p}$-gradient Young measures $\nu$, where $\bar{\nu} := \int_{\mathbb{R}^{m\times n}} \lambda d\nu(\lambda)$ denotes the average.
A lesser known concept is that of quasimonotonicity, which seems to occur mainly in a series of works by Hungerbühler and coauthors [1,2], who use it to prove existence of weak solutions to certain PDEs: A function $\eta:\mathbb{R}^{m\times n} \to \mathbb{R}^{m\times n}$ (again with reasonable growth conditions) is called (strictly) quasimonotone if $$\int_{\mathbb{R}^{m\times n}} (\eta(\lambda)-\eta(\bar{\nu})): (\lambda-\bar{\nu}) d\nu(\lambda) \geq 0 $$ for all homogeneous $W^{1,p}$-gradient Young measures $\nu$ (nonzero whenever $\nu$ is not a Dirac measure). There are some minor variants of that definition, but the idea is always more or less the same. (Though, as with quasiconvexity, there are also other unrelated definitions sharing the same name.)
It is a classical result that gradients of convex functions are monotone operators. (In fact gradients of convex functions can be characterized as cyclical monotone operators, which is a slightly stronger condition). In contrast, both [1] and [2] state that not all gradients of quasiconvex functions result in quasimonotone operators. However neither provides more detailed explanation or a reference for that statement. Thus the following question:
What is the relationship between the gradient of quasiconvex functions and quasimonotone operators? Is there a general counterexample or can this be saved under some minor extra conditions? In particular also, has this notion of quasimonotonicity been studied in more detail somewhere?
- [1] Dolzmann, Georg, Norbert Hungerbühler, and Stefan Müller. 1997. ‘Non-Linear Elliptic Systems with Measure-Valued Right Hand Side’. Math Z. 226: 545–74.
- [2] Hungerbühler, Norbert. 1999. ‘Quasilinear Elliptic Systems in Divergence Form with Weak Monotonicity’. New York J. Math 5 (83): 90.
