for some $R_K>0$ depending only on $(K,d,\kappa,\delta,\theta)$.

Our assumptions allow for the previous example to be generalized to the anisotropic setting. We may consider the case that $$A(\xi,x)=\left(I_d- \frac{\xi}{\left| \xi \right|}\otimes \frac{\xi}{\left| \xi \right|} \right){\widetilde}A(\hat \xi,x)\left(I_d- \frac{\xi}{\left| \xi \right|}\otimes \frac{\xi}{\left| \xi \right|}\right),  \qquad H(\xi,x)= \left|B\left( \frac{\xi}{\left| \xi\right|},x\right)\xi\right|,$$ where ${\widetilde}A$ takes the form ${\widetilde}A= \frac12 {\widetilde}\sigma{\widetilde}\sigma^T$ and $B$ is a matrix-valued random field satisfying $c_0I_d \leq B^TB \leq C_0 I_d$. Note that in this context $\sigma= \left(I_d-\hat \xi\otimes \hat \xi\right){\widetilde}\sigma$. Then the following condition implies that the (LS) condition holds: $$\inf_{(e,x) \in \partial B_1\times {\mathbb{R}^d}} \left[\left|B(e,x)e\right| -| \sigma(e,x)|^2\left(| \sigma_x(e,x)|^2+|B(e,x)e||B_x(e,x)|\right)\right] >0.$$ We leave the confirmation of this claim to the reader.

Brief review of the literature
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Periodic and almost-periodic homogenization results were proved in [@LS0] for the type of quasilinear and geometric Hamilton-Jacobi equations considered here. The fundamental qualitative homogenization results for convex Hamilton-Jacobi equations in the random setting were first proved in [@RT; @S] in first-order case and later in the semilinear, viscous case in [@KRV; @LS]. New proofs and extensions of these results appeared later in [@LS3; @Sch; @AS1; @AS2].

As explained above, for geometric equations, the existence of plane-like solutions in periodic media was proved in [@CD]; see 