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I need to evaluate the following limit which intuitively I know its equal to 0 but I can't really prove it, so I need some help:

$$\lim_{\epsilon \to 0}{\frac{F[\rho + \epsilon\rho' + \epsilon^2\rho'']-F[\rho + \epsilon\rho']}{\epsilon}}$$

where $\epsilon$ is a real number, $F$ is a functional and $\rho$, $\rho'$ and $\rho''$ are functions in some function space.

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What is the functional $F$? Or are you supposed to prove that the limit is zero for all possible functionals? –  Rod Carvalho Sep 12 '12 at 4:36
    
I don't know the form of the functional, except that it is local. I started trying to prove a general identity involving functionals and I ended with this limit. Probably it needs some additional assumptions. –  Manuel Sep 12 '12 at 4:49
    
Is $F$ supposed to be linear? –  Christopher A. Wong Sep 12 '12 at 4:53
    
No, linearity is not assumed. –  Manuel Sep 12 '12 at 5:05
    
Does $F$ obey any sublinearity properties, or are there any inequalities given for $F$? –  Christopher A. Wong Sep 12 '12 at 5:08
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1 Answer

up vote 1 down vote accepted

Here is a very, very rough sketch of a possible proof...

Let $\bar{\rho} (\epsilon) := \rho + \epsilon \rho'$. The limit can then be written in the form

$$\displaystyle\lim_{\epsilon \to 0} \frac{1}{\epsilon}\left[ F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') - F (\bar{\rho} (\epsilon))\right]$$

The "Taylor expansion" of $F$ is the following

$$F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') = F (\bar{\rho} (\epsilon)) + \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon^2 \rho''\rangle + \omicron (\epsilon^4)$$

where $\nabla F (\bar{\rho} (\epsilon))$ is the "functional gradient" of $F$. Then, we have that

$$\displaystyle\lim_{\epsilon \to 0} \frac{1}{\epsilon}\left[ F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') - F (\bar{\rho} (\epsilon))\right] = \lim_{\epsilon \to 0} \left[\frac{1}{\epsilon} \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon^2 \rho''\rangle + \omicron (\epsilon^3)\right]$$

If we can show that

$$\frac{1}{\epsilon} \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon^2 \rho''\rangle = \langle \nabla F (\bar{\rho} (\epsilon)), \epsilon \rho''\rangle$$

then the limit becomes

$$\displaystyle\lim_{\epsilon \to 0} \frac{1}{\epsilon}\left[ F (\bar{\rho} (\epsilon) + \epsilon^2 \rho'') - F (\bar{\rho} (\epsilon))\right] = \lim_{\epsilon \to 0} \left[\langle \nabla F (\bar{\rho} (\epsilon)), \epsilon \rho''\rangle + \omicron (\epsilon^3)\right] = 0$$

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This is ok for my purposes, I also figured some kind of Taylor expansion for functionals but I didn't know if something like that really existed. Can you point me to literature where I can find the theory behind these expansions? Thank you. –  Manuel Sep 12 '12 at 18:02
    
@Manuel: This is a good introductory overview of calculus of variations: math.umn.edu/~olver/am_/cvz.pdf –  Rod Carvalho Sep 12 '12 at 19:43
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