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Some homogenization results for non-coercive Hamilton-Jacobi equations. (English) Zbl 1136.35004
The author studies the limit behaviour as \(\varepsilon \to 0\) of the Hamilton-Jacobi equations
\[ U_t^{\varepsilon} + F({\varepsilon}^{-1} x, {\varepsilon}^{-1} y, {\varepsilon}^{-1}t, D_x U^{\varepsilon}, D_yU^{\varepsilon}) = 0 \quad \text{in }{\mathbb R}^{n+1} \times (0,+\infty) \] where \(x \in {\mathbb R}^{n}\), \(y \in {\mathbb R}\), \(t \in (0,+\infty)\) and \(F(x,y,t,p_x,p_y)\) is a function continuous in \({\mathbb R}^{2n+3}\) and \({\mathbb Z}^n\)-periodic in \(x\) and \(1\)-periodic in \(y,t\). Usually \(F\) is supposed to be coercive with respect to the variables \((p_x,p_y)\) and this is useful in solving the cell problem which provides the limit problem.
The author studies homogenization dropping coerciveness in \((p_x,p_y)\). He also obtains another proof for a recent result of C. Imbert and R. Monneau [Arch. Ration. Mech. Anal. 187, No. 1, 49-89 (2008; Zbl 1127.70009)], the paper that inspired the author for the present work.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35F20 Nonlinear first-order PDEs
35F25 Initial value problems for nonlinear first-order PDEs
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
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