Imagine variational calculus on nonlinear multifunction functionals. Let $C^k[a,b]$ denote the set of all $k$ times differentiable functions from $[a,b]$ to $\mathbb{R}$. Now consider a functional $F[\mu,\lambda]:C^k[a,b] \times C^k[a,b] \to \mathbb{R}$, and the extrema of this functional.
So, for example, one class of functional that arose of interest to me in this area was $$\mathbb{D}[x,y] = \int_0^1 \sqrt{\left(\frac{\text{d}x(\ell)}{\text{d} \ell} \right)^2 + \left( \frac{\text{d}y(\ell)}{\text{d}\ell} \right)^2} \ \text{d}\ell + k_{\omega} \left(1 - \exp{\left(\int_0^1 \ln{\left( \iint_{\Omega} \psi\left(\left \langle \begin{matrix} x(\ell) \\ y(\ell) \end{matrix} \right \rangle,\left \langle \begin{matrix} x_\pi \\ y_\pi \end{matrix} \right \rangle \right) \sigma\left(\left \langle \begin{matrix} x_\pi \\ y_\pi \end{matrix} \right \rangle \right) \ \text{d}x_\pi \ \text{d}y_\pi \right)} \ \text{d}\ell \right) } \right) $$ where $x(\ell)$ and $y(\ell)$ are any real valued functions from the unit interval to the numberline, and $\psi$ and $\sigma$ are positive real-valued functions. This is example isn't strictly relevant to the question but it may offer some insight into what I'm asking about.
In the most general form, for a functional $F$ like that written above, I think any extremizing pair of functions $\mu$ and $\lambda$ must be unimprovable by small variations in both directions, requiring that for any pair of functions $\xi$ and $\eta$ vanishing on the boundary, so I'm pretty sure that it's necessary for any extremizers to satisfy
$$\forall \xi, \eta: \lim_{(\varepsilon_\mu, \varepsilon_\lambda) \to(0,0)} \frac{F[\mu + \varepsilon_\mu \xi, \lambda + \varepsilon_\lambda \eta]-F[\mu, \lambda]}{\sqrt{\varepsilon_\mu^2 + \varepsilon_\lambda^2}} = 0 $$
(I should note that I'm not sure this is the best limit definition. The denominator may be in fact something like $\varepsilon_\mu \varepsilon_\lambda$). Now, within non linearity of the functional I think it's possible this is not equivalent to the simultaneous conditions $$\forall \xi: \lim_{\varepsilon \to 0} \frac{F[\mu + \varepsilon \xi, \lambda]-F[\mu, \lambda]}{\varepsilon} $$ and $$\forall \eta: \lim_{\varepsilon \to 0} \frac{F[\mu , \lambda+\varepsilon \eta]-F[\mu, \lambda]}{\varepsilon} $$ but I think it's possible that I can, for most functionals at least, rewrite this as
$$\forall \xi, \eta: \lim_{\varepsilon\to0} \frac{F[\mu + \varepsilon \xi, \lambda + \varepsilon \eta]-F[\mu, \lambda]}{\varepsilon\sqrt2} = 0 $$
of course meaning that the limit does in fact exist and is the same for all paths. So, my question is this:
1. What is the best limit definition of the functional derivative for multifunction functionals?
and
2. For what classes of functions can we simplify this limit definition, especially by rewriting it as a pair of simultaneous conditions each varying one of the functions?