2023/09/17 by Elena Kosygina, Kosygina, Elena, Atilla Yılmaz +1
Mathematics · #35B27 (Primary) 35F21 #35D40 (Secondary) #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2309.09343
openalex publication_date 2023/09/17 · openalex created_date 2023/09/20 · openalex updated_date 2026/07/28
We show that, in the periodic homogenization of uniformly elliptic Hamilton-Jacobi equations in any dimension, the effective Hamiltonian does not necessarily inherit the quasiconvexity property (in the momentum variables) of the original Hamiltonian. This observation is in sharp contrast with the first order case, where homogenization is known to preserve quasiconvexity. We also show that the loss of quasiconvexity is, in a way, generic: when the spatial dimension is 1, every convex function G can be modified on an arbitrarily small open interval so that the new function G is quasiconvex and, for some 1-periodic and Lipschitz continuous V, the effective Hamiltonian arising from the homogenization of the uniformly elliptic Hamilton-Jacobi equation with the Hamiltonian H(p,x)=G(p)+V(x) is not quasiconvex.