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August 16, 2026SIAM Journal on Mathematical AnalysisOpen Access

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

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Authors

EKElena KosyginaAYAtilla Yılmaz

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Overview

Mathematical analysis demonstrates that periodic homogenization of viscous Hamilton–Jacobi equations fails to preserve quasiconvexity, indicating fundamental divergence from first-order systems.

Key Points

  • To determine whether quasiconvexity of Hamiltonians is preserved under periodic homogenization of uniformly elliptic viscous Hamilton–Jacobi equations across arbitrary dimensions.
  • Analyzed the periodic homogenization of uniformly elliptic Hamilton–Jacobi equations with quasiconvex Hamiltonians in arbitrary spatial dimensions.
  • Constructed explicit one-dimensional perturbations of convex Hamiltonians on small intervals combined with 1-periodic Lipschitz continuous functions.
  • Demonstrated that the effective Hamiltonian resulting from periodic homogenization of uniformly elliptic equations is not necessarily quasiconvex, contrasting sharply with first-order systems.
  • Established that the loss of quasiconvexity is generic in one dimension, where any convex function can be locally modified to yield a non-quasiconvex effective Hamiltonian.

Cite This Study

Kosygina et al. (2026) studied this question.

synapsesocial.com/papers/6a8178fcf2fb91fc834ac239https://doi.org/10.1137/25m1780055
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Also Consider

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  1. 1Stochastic homogenization of quasiconvex degenerate viscous HJ equations in 1d2024
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  3. 3Stochastic Homogenization of a Class of Quasiconvex and Possibly Degenerate Viscous HJ Equations in 1D2024 · 1 citations
  4. 4Optimal convergence rate for homogenization of convex Hamilton–Jacobi equations in the periodic spatial-temporal environment2024 · 5 citations
  5. 5Homogenization of nonconvex viscous Hamilton-Jacobi equations in stationary ergodic media in one dimension2024