Harmonic Analysis and Elliptic Homogenization Problems

The P1 proposes to initiate the study of a class of elliptic homogenization problems in domains with non-smooth boundaries. The main focus will be on the second order elliptic systems with rapidly oscillating periodic coefficients in Lipschitz and C' do- mains. The primary objective of this project is to gain a better understanding of the boundary regularity properties of solutions by establishing uniform estimates under physically realistic assumptions. Intellectual Merit: Boundary value problems in non-smooth domains with L~ data have received extensive study in the last 30 years. However, very few results are known for elliptic equations and systems with rapidly oscillating periodic coefficients, which arise in the theory of homogenization. The dilation-invariant property of the family of elliptic operators {E e > 0} and that of the class of Lipschitz domains make problems to be investigated very interesting and challenging. Recently, motivated by the work of Caffarelli and Peral, the P1 developed a new approach to study the I? problems. Starting with L2 estimates, the new real variable approach allows one to prove the corresponding L" estimates for 2 cc p < q, if certain weak reverse Holder type inequalities with exponent q can be established in every scale. This fairly general method is particularly effective in the non-smooth setting, where the L~ estimates are not always expected to hold for all 2