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We study a cell problem arising in homogenization for a Hamilton-Jacobiequation whose Hamiltonian is not coercive. We introduce a generalized notionof effective Hamiltonians by approximating the equation and characterize thesolvability of the cell problem in terms of the generalized effectiveHamiltonian. Under some sufficient conditions, the result is applied to theassociated homogenization problem. We also show that homogenization fornon-coercive equations fails in general.

On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems