2
$\begingroup$

Let $X$ be a infinite dimensional Banach space and $X^\star$ its dual. Let $f:X\rightarrow X^\star$ be a continuous function. What is a necessary and sufficient condition to find $F:X\rightarrow\mathbb{R}$ such that $F'(x)=f(x)$ where $F'$ is the derivative of $F$.

Note: In Banach spces with finite dimension we have the notion of "conservative field". What I am asking here is the analogous in infinite dimesion.

1 Answers 1

2

A classical theorem of Rockafellar characterizes the gradients of convex functions.

Theorem B. Let $T:E\to E^*$ be a multivalued mapping. In order that there exist a lower semicontίnuous proper convex function $f$ on $E$ such that $T=\partial f$, it is necessary and sufficient that $T$ be a maximal cyclically monotone operator. Moreover, in this case $T$ determines $f$ uniquely up to an additive constant.

Since your $f$ is continuous, you can ignore "multivalued" and "maximal". As a corollary, if $f$ can be written as the difference of two cyclically monotone maps then it is the gradient (of the difference of two corresponding functions). A function that can be written as the difference of two convex functions is called delta-convex (or d.c.). This is a pretty large class; in finite dimensions in includes all $C^2$ functions. There is a fair amount of recent literature on d.c. functions, but I don't remember seeing any better characterization of their gradients.

  • 0
    Thank you @Tomás for posing interesting problems. :) But you did not have to accept my answer; there's at least a chance someone knows something about non-convex case.2012-12-18