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We study two types of asymptotic problems whose common feature - anddifficulty- is to exhibit oscillating Dirichlet boundary conditions : the maincontribution of this article is to show how to recover the Dirichlet boundarycondition for the limiting equation. These two types of problems are (i)periodic homogenization problems for fully nonlinear, second-order ellipticpartial differential equations set in a half-space and (ii) parabolic problemswith an oscillating in time Dirichlet boundary condition. In order to obtainthe Dirichlet boundary condition for the limiting problem, the key step is ablow-up argument near the boundary which leads to the study of Dirichletproblems set on half space type domains and of the asymptotic behavior of thesolutions when the distance to the boundary tends to infinity.

On homogenization problems for fully nonlinear equations with oscillating Dirichlet boundary conditions