Hayk Aleksanyan: Convergence rates in periodic homogenization of Dirichlet problem for divergence type elliptic systems

We consider Dirichlet problem for divergence type elliptic systems with fixed operator and periodically oscillating boundary data. For smooth and uniformly convex domains we prove pointwise, as well as L^p convergence results for this homogenization problem. In addition, we prove that the obtained L^p convergence rate is generically sharp in dimensions greater than 3. For some class of operators, we combine our techniques with a recent result due to C. Kenig, F. Lin and Z. Shen, to obtain homogenization in L^p with optimal rate of convergence, in case when oscillations are present both in operator and in boundary data. We will also discuss similar results for polygonal domains, under certain Diophantine condition on the normals of the bounding hyperplanes of the domain. This is a joint work with Henrik Shahgholian and Per Sjölin.