Loss of Quasiconvexity in the Periodic Homogenization of Viscous Hamilton–Jacobi Equations

We show that in the periodic homogenization of uniformly elliptic Hamilton–Jacobi equations with quasiconvex Hamiltonians in arbitrary dimensions, the effective Hamiltonian is not necessarily quasiconvex. This observation is in sharp contrast with the first-order case, where homogenization is known to preserve quasiconvexity, even within a much broader class of Hamilton–Jacobi equations in stationary ergodic environments. We also demonstrate that the loss of quasiconvexity is, in a certain sense, generic: when the spatial dimension is one, any convex function [Formula: see text] can be modified on an arbitrarily small open interval so that the new function [Formula: see text] is quasiconvex, and for some 1-periodic and Lipschitz continuous function [Formula: see text], the effective Hamiltonian resulting from the homogenization of the uniformly elliptic Hamilton–Jacobi equation with Hamiltonian [Formula: see text] is not quasiconvex.