My question is the following:
Given a vector field $\mathbf{v}$, we have the following functional: $f\{\mathbf{v}\}=1/|\mathbf{v}|_{\text{max}}$, where $|\mathbf{v}|_\text{max}$ is the maximum of the 2-norm of the vector field $\mathbf{v}$ over $\Omega$ (where $\Omega$ is the domain of interest, which is bounded). Now lets assume we perturb the vector field slightly: $\mathbf{v}\rightarrow\mathbf{v}+\Delta{\mathbf{v}}$, where $|\Delta \mathbf{v}|\ll |\mathbf{v}|_{\text{max}}$ over $\Omega$. To lowest order, $f\{\mathbf{v+\Delta\mathbf{v}}\}\sim1/|\mathbf{v}|_\text{max}$ .
My question is: what would the first order correction term be (i.e. first order in $\Delta\mathbf{v}$)??
(Additional assumptions include that both $\mathbf{v}$ and $\Delta\mathbf{v}$ are smooth and continuous, and remain bounded in $\Omega$)
Thanks.
I think I have found an answer to my own question. Thoughts/comments would be appreciated however.
So if we assume that $\Delta \mathbf{v}$ is sufficiently small in magnitude, (but still an arbitrary perturbation), then we can expand the max functional to first order in $\Delta \mathbf{v}$.
We first define the max functional as $$\text{max}\{f\}\equiv\cfrac{\underset{\Omega}{\int} \text{d}^3 x fe^{\alpha f}}{\underset{\Omega}{\int} \text{d}^3 x e^{\alpha f}}\ ,\tag{1}\label{1}$$ where $\alpha$ is a very large number. We will also for convenience define $G\{f\}\equiv \underset{\Omega}{\int} \text{d}^3 x fe^{\alpha f}$ and $H\{f\}\equiv \underset{\Omega}{\int} \text{d}^3 x e^{\alpha f}$. In other words, $$\text{max}\{f\}=\cfrac{G\{f\}}{H\{f\}} \tag{2}\label{2}\ .$$.
If we expand both $G\{f\}$ and $H\{f\}$, each as a zeroth-order and first-order contribution in $\Delta \mathbf{v}$, and then expand the denominator of Eq. (\ref{2}), Eq. (\ref{2}) becomes: $$\text{max}\{f\}\approx\cfrac{G_0}{H_0}+\cfrac{G_1}{H_0}-\cfrac{G_0H_1}{H_0^2}\ ,\tag{3}\label{3}$$ where subscripts refer to the order of the terms in the expansions for $G$ and $H$ respectively. All we then have left to do is find the first order contributions to $G$ and $H$ (i.e. $G_1$ and $H_1$).
We first note that $|\mathbf{v}+\Delta \mathbf{v}|\approx |\mathbf{v}|+(\mathbf{v}\cdot\Delta \mathbf{v})/{|\mathbf{v}}|$ , where we have expanded the input to the functionals into a zeroth-order term and a first-order term. Using this, we Taylor-expand the integrand in each functional about $\mathbf{v}$: $$G_1\{\mathbf{v},\Delta \mathbf{v}\}=\underset{\Omega}{\int}\text{d}^3 x (1+\alpha |\mathbf{v}|)e^{\alpha |\mathbf{v}|}\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|}\right)\ ,$$ and $$H_1\{\mathbf{v},\Delta \mathbf{v}\}=\underset{\Omega}{\int}\text{d}^3 x \alpha e^{\alpha |\mathbf{v}|}\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|}\right)\ .$$ If we define the $x$ value at which the maximum of $|\mathbf{v}|$ occurs to be $x_{\text{max}}$, then it is easily seen that: $$\cfrac{G_0}{H_0}=|\mathbf{v}|_\text{max}\ ,$$ $$\cfrac{G_1}{H_0}=\left.\left[(1+\alpha |\mathbf{v}|)\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|}\right)\right]\right|_{x_\text{max}}\ ,$$ and $$-\cfrac{G_0H_1}{H_0^2}=-|\mathbf{v}|_\text{max}\alpha\left.\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|}\right)\right|_{x_{\text{max}}}\ .$$
Plugging the above three results into Eq. (\ref{3}), we obtain that $$\text{max}\{|\mathbf{v}+\Delta \mathbf{v}|\}\approx |\mathbf{v}|_\text{max}+\left.\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|}\right)\right|_{x_\text{max}}\ .$$ From this, we obtain the final result: $$\cfrac{1}{\text{max}\{|\mathbf{v}+\Delta \mathbf{v}|\}}\approx \cfrac{1}{|\mathbf{v}|_\text{max}}-\left.\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|^3}\right)\right|_{x_\text{max}}\ .$$
This is a linear expansion, because $x_\text{max}$ does not depend on $\Delta \mathbf{v}$.
This analysis assumes that there is only one $x_\text{max}$ . If there are multiple $x_\text{max}$, and using the same above analysis, it can be shown that $$\cfrac{1}{\text{max}\{|\mathbf{v}+\Delta \mathbf{v}|\}}\approx \cfrac{1}{|\mathbf{v}|_\text{max}}-\left(\cfrac{\mathbf{v}\cdot\Delta \mathbf{v}}{|\mathbf{v}|^3}\right)_\text{ave}\ ,$$ where $()_\text{ave}$ is the average value of the quantity over all the $x_\text{max}$ in $\Omega$.