Applications of Fourier analysis in homogenization of Dirichlet problem I. Pointwise Estimates

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, namely $$|u_{\e}(x)-u_0 (x)| \leq C_{\kappa} \e^{(d-1)/2}\frac{1}{d(x)^{\kappa}}, \ \forall x\in D, \ \forall \ \kappa>d-1,$$ where $u_{\e}$ 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 $1\leq p <\infty$ we obtain $L^p$ convergence rate as well.