In this paper we prove convergence results for homogenization problem for solutions of partial differential system with rapidly oscillating Dirichlet data. Our method is based on analysis of oscillatory integrals. In the uniformly convex and smooth domain, and smooth operator and boundary data, we prove pointwise convergence results, namelywhere u ε and u 0 are solutions of respectively oscillating and homogenized Dirichlet problems, and d(x) is the distance of x from the boundary of D. As a corollary for all 1 ≤ p < ∞ we obtain L p convergence rate as well.

Abstract. In this paper we prove convergence results for the homogenization of the Dirichlet problem with rapidly oscillating boundary data in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as L p convergence results. For larger exponents p we prove that the L p convergence rate is close to optimal. We shall also suggest several directions of possible generalization of the result in this paper.

Abstract. In this paper we prove convergence results for the homogenization of the Dirichlet problem with rapidly oscillating boundary data in convex polygonal domains. Our analysis is based on integral representation of solutions. Under a certain Diophantine condition on the boundary of the domain and smooth coefficients we prove pointwise, as well as L p convergence results. For larger exponents p we prove that the L p convergence rate is close to optimal. We shall also suggest several directions of possible generalization of the result in this paper.

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