I am interested in a family of PDE which have defeated my (admittedly rather naive) attempts to prove any regularity or stability estimates. These are systems of PDE which are elliptic in some directions, degenerate in others, and moreover the degenerate directions are allowed to depend on the solution in a nonlinear fashion.
To be more precise, let $2 < p < \infty$, and for a matrix $A$, let $Q(A) := \sqrt{AA^\dagger}$ be the positive-semidefinite part of $A$. The PDE in question is $$\nabla \cdot (Q(\nabla u)^{p - 2} \nabla u) = 0$$ (where $u: \Omega \to \mathbb R^D$ for $\Omega \Subset \mathbb R^d$ a nice domain) which arises as the analogue of the $p$-Laplacian when one tries to generalize the $\infty$-Laplacian to the optimal Lipschitz maps problem; see Daskalopoulos and Uhlenbeck's "Analytic properties of stretch maps and geodesic laminations".
By ideas already in "Analytic properties" (nothing fancier than pointwise inequalities on matrices and integration by parts!) one can show that if $d = D = 2$ then $\nabla u \in BMO$. So it is natural to conjecture that by higher-tech means one could show $\nabla u \in L^\infty$. However, this is subtle. To prove $\nabla u \in L^\infty$ for the scalar $p$-Laplacian, one can break into cases $|\nabla u| \geq 1$ and $|\nabla u| \leq 1$ -- in the former case the equation is elliptic and in the latter case there is nothing to prove. But related ideas presumably fail here, because one can have $|\nabla u| \gg 1$ and yet $Q(\nabla u)^{p - 2}$ can be degenerate. A nice way to see this is to take $p = 4$ and linearize around $u(x, y) = (x, 0)$; the linearized equation is $$\begin{bmatrix}3 \partial_{xx} v_1 + \partial_{yy} v_1 \\ \partial_{xx} v_2 \end{bmatrix} = 0$$ so that, on the linear level, we have no control on $\partial_y v_2$.
While I guess that this particular PDE has not been considered before, surely the idea that the degeneracy of a PDE should depend on its solution is not new. In fact, it can be seen in the scalar $\infty$-Laplacian: when $d = 2$ and $D = 1$, solutions of $$\langle \nabla^2 u, \nabla u \otimes \nabla u\rangle = 0$$ are degenerate in the direction orthogonal to $\nabla u$ but satisfy $\nabla u \in C^\alpha$ by a tricky argument of Evans and Savin -- but I don't think this is terribly helpful for systems, since it is based on comparison techniques.
So that's what I'd like to ask:
Consider systems of PDE, which are degenerate in some directions but elliptic in others, but still enjoy nontrivial estimates establishing regularity or stability. What are examples of such PDE? What techniques are used to study them?
