Plane-like solutions are intimately connected to homogenization; it is not hard to see that their existence for every given slope implies homogenization for general initial data (at least in the context in which one has Lipschitz solutions for sufficiently smooth initial data, which is assured by our hypotheses). In the random setting, we do not expect that plane-like solutions exist, in general. Indeed, their existence would imply a rate of homogenization of $O({\varepsilon})$ for affine initial data, while in dimension $d=2$ we expect a convergence rate of $O({\varepsilon}^{\frac23})$, in line with the conjectured bound for first-passage percolation (a discrete version of a first-order Hamilton-Jacobi equation) and other growing surface models expected to possess a scaling limit related to the KPZ equation.

Our strategy for homogenization is nevertheless based on the construction of a weaker variant of a plane-like solution in half spaces. We consider the problem posed in the half-space ${\mathcal{H}}_e^+ = \{ x\cdot e>0\}$, for a given parameter $\mu>0$ and unit vector $e$: $$\label{}
\left\{ 
\begin{aligned}
&  -\operatorname{tr}\left( A(Dm_\mu,x) D^2m_\mu\right) + H(Dm_\mu,x) = \mu & \mbox{in} & \ {\mathcal{H}}_e^+, \\
& m_\mu = 0 & \mbox{on} & \ \partial {\mathcal{H}}_e^+
\end{aligned}
\right.$$ We call this the *planar metric problem*, 