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

Abstract

We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general.

Two approaches to minimax formula of the additive eigenvalue for quasiconvex Hamiltonians

“…Hence, H n becomes coincident to H on D + u for sufficiently large n. Therefore, c(H n ) = c + (H n ) ≤ a, which shows lim sup n c(H n ) ≤ c + (H). For general approximations one can show by the same arguments as in [17,Theorem 4.1] that c(H n ) is a convergent sequence and that the limit does not depend on the choice of the approximations. Finally, we have lim n c(H n ) = c + (H).…”

“…The proof is trivial so we omit it. This is a well-known fact; we refer the reader to [18], [14] and [17]. Under the assumptions (A1) and (A3) the unique eigenvalue c = c(H) := c + (H) = c − (H) is called critical value of (1.1).…”

“…The generalized effective Hamiltonian introduced in the author's previous work [17] is nothing but the upper critical value c + : Proposition 2.4. Assume that H satisfies (A1) and let H n be a sequence of Hamiltonians satisfying (A1) and (A3).…”

“…This Hamiltonian is quasiconvex (A2') as well as non-coercive. Long time behavior and homogenization for this Hamiltonian have been studied in [15] and [17]. In order to explain the main idea of one of the proofs, let us review the known proof under the assumptions (A1) and (A2).…”

“…This Hamiltonian is quasiconvex (A2') as well as non-coercive. Long time behavior and homogenization for this Hamiltonian have been studied in [15] and [17].…”