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Abstract We study the effective front associated with first-order front propagations in two dimensions ($n=2$) in the periodic setting with continuous coefficients. Our main result says that that the boundary of the effective front is differentiable at every irrational point. Equivalently, the stable norm associated with a continuous $${\mathbb{Z}}^{2}$$-periodic Riemannian metric is differentiable at irrational points. This conclusion was obtained decades ago for smooth metrics [ 4, 6]. To the best of our knowledge, our result provides the first nontrivial property of the effective fronts in the continuous setting, which is the standard assumption in the literature of partial differential equations (PDE). Combining with the sufficiency result in [ 15], our result leads to a realization type conclusion: for continuous coefficients, a polygon could be an effective front if and only if it is centrally symmetric with rational vertices and nonempty interior. 

G-equations are popular level set Hamilton–Jacobi nonlinear partial differential equations (PDEs) of first or second order arising in turbulent combustion. Characterizing the effective burning velocity (also known as the turbulent burning velocity) is a fundamental problem there. We review relevant studies of the G-equation models with a focus on both the existence of effective burning velocity (homogenization), and its dependence on physical and geometric parameters (flow intensity and curvature effect) through representative examples. The corresponding physical background is also presented to provide motivations for mathematical problems of interest. Thelack of coercivityof Hamiltonian is a hallmark of G-equations. When either the curvature of the level set or the strain effect of fluid flows is accounted for, the Hamiltonian becomeshighly nonconvex and nonlinear. In the absence of coercivity and convexity, the PDE (Eulerian) approach suffers from insufficient compactness to establish averaging (homogenization). We review and illustrate a suite of Lagrangian tools, most notably min-max (max-min) game representations of curvature and strain G-equations, working in tandem with analysis of streamline structures of fluid flows and PDEs. We discuss open problems for future development in this emerging area of dynamic game analysis for averaging noncoercive, nonconvex, and nonlinear PDEs such as geometric (curvature-dependent) PDEs with advection.