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Homogenization of first-order equations with \((u/\varepsilon)\)-periodic Hamiltonians. I: Local equations. (English) Zbl 1127.70009
Summary: We present a result on homogenization of first-order Hamilton-Jacobi equations with \((u/\varepsilon)\)-periodic Hamiltonians. On the one hand, under a coercivity assumption on the Hamiltonian (and some natural regularity assumptions), we prove an ergodicity property of this equation and the existence of nonperiodic approximate correctors. On the other hand, the proof of convergence of the solution, usually based on the introduction of a perturbed test function in the spirit of L. C. Evans’s work [Proc. R. Soc. Edinb., Sect. A 111, No. 3/4, 359–375 (1989; Zbl 0679.35001)], uses here a twisted perturbed test function for a higher-dimensional problem.

MSC:
70H20 Hamilton-Jacobi equations in mechanics
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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