Let $Pu = f$ be an elliptic PDE in divergence form. Then $P$ is viewed as a generalization of the Laplacian, and we can define its weak solutions analogously to how we define a weakly harmonic function: just integrate by parts to put the divergence on a test function. This procedure also works for certain systems of elliptic PDE. Ignoring issues of gauge-invariance for the sake of example, we can rewrite the magnetostatic Maxwell equation $d^* dA = 0$ as $$\int_\Omega \langle dA, d \psi\rangle dV = 0$$ for every $1$-form $\psi$ of regularity $C^\infty_c(\Omega, \mathfrak g)$.
For $p \in [1, \infty]$, let me informally call an equation "$p$-elliptic" if it is a generalization of the $p$-Laplacian in some sense. In particular, it should arise as the Euler-Lagrange equation for the minimization of a suitable $L^p$ norm. For $p \in (1, \infty)$, the $p$-Laplacian is in divergence form $$\nabla \cdot (|\nabla u|^{p - 2} \nabla u) = 0$$ and again we can use the integration by parts trick. Again this works for systems: the magnetostatic $p$-Maxwell equation for a $1$-form $A$, $$d^* (|dA|^{p - 2} dA) = 0$$ which is the Euler-Lagrange equation of $\|dA\|_{L^p}$, is in divergence form. Things get a little hairy when $p = 1$, but it's still in divergence form, so as shown by Mazón, Rossi, and Segura de León in their paper "Functions of least gradient and $1$-harmonic functions", you can still define weakly $1$-harmonic functions (in a way that seems to extend nicely to certain systems).
My question is about the other endpoint $p = \infty$. Recall that the $\infty$-Laplacian is $$\nabla^i \partial^j u \partial_i u \partial_j u = 0$$ which is not in divergence form. There are two good notions of weak solution here: viscosity solutions, and comparison with cones. They turn out to be equivalent, and both essentially amount to asserting that $u$ satisfies the correct form of the maximum principle. In particular, $u$ must be a scalar field for something like comparison with cones or viscosity solutions to make sense.
So integration by parts, viscosity solutions, and comparison with cones don't seem to apply to the Euler-Lagrange equations for absolute minimizers of $\|dA\|_{L^\infty}$ (among other systems). Using the machinery of Barron, Jensen, and Wang's "Lower semicontinuity of $L^\infty$ functionals" paper one can formally deduce that the Euler-Lagrange equations "should be" $$(dA)^{ij} \partial_i |dA| = 0.$$ I will call this equation the "$\infty$-Maxwell equation" for the sake of discussion. For readers more comfortable with the vector field formalism, I note that in dimension $3$, if we introduce the dual vector field $X = A^\sharp$, the $\infty$-Maxwell equation is equivalent to $$(\nabla \times X) \times \nabla |\nabla \times X| = 0.$$ Notice that when you integrate by parts to move the second derivative onto a test function, it partially becomes a divergence on the $F^{ij}$ (equivalently $\nabla \times X$) instead!
Anyways, it's not at all clear to me how one would ever hope to study this equation. You can establish existence of solutions by $p$-approximation, similarly to how Daskalopolous and Uhlenbeck construct $\infty$-harmonic maps in their papers "Best Lipschitz and least gradient maps and transverse measures" and "Analytic properties of stretch maps and geodesic laminations", but without comparison with cones, you lose everything good about the $\infty$-Laplacian. There's no clear way to establish any sort of uniqueness (say, in Coulomb gauge $d^* A = 0$) or that solutions should be absolute minimizers. You probably don't get any regularity (other than trivially that Coulomb gauge solutions are Lipschitz). Actually, in "Analytic properties", the authors are interested in $\mathbf H^2$-valued $\infty$-harmonic maps, but give up on defining weak solution for that equation; they have to prove everything by working with $p$-harmonic maps and taking weak limits, which is an approach with strong limitations.
So we need a notion of weak solution to seriously study the $\infty$-Maxwell equation. Thus:
What is the correct notion of weak solution for the $\infty$-Maxwell equation, and others like it?
Here's something I haven't seriously tried: Write each out component of the $\infty$-Maxwell equation separately, and hope that it looks like a system of individual $\infty$-Laplacians, which are sufficiently decoupled (possibly up to a gauge transformation?) that one can think about viscosity solutions of each $\infty$-Laplacian separately. Anyways, I'm mainly asking to see if there's a known approach that I'm missing, which avoids this "reduction to the $\infty$-Laplacian" which seems kind of intractable.
