Bounds of Riesz Transforms on LpL^p Spaces for Second Order Elliptic Operators

For any fixed $p>2$, a necessary and sufficient condition is obtained for the boundedness of the Riesz transforms associated with second order elliptic operators with real, symmetric, bounded measurable coefficients.Comment: To appear in Annales de L'Institut Fourier (Grenoble

The L^p Boundary Value Problems on Lipschitz Domains

We develop a new approach to the invertibility of the layer potentials on $L^p$ associated with elliptic equations and systems in Lipschitz domains. As a consequence, for $n\ge 4$ and $(2(n-1)/(n+1))-\epsilon<p<2$, we obtain the solvability of the L^p Neumann type boundary value problems for second order elliptic systems. The analogous results for the biharmonic equation are also established

Convergence Rates and H\"older Estimates in Almost-Periodic Homogenization of Elliptic Systems

For a family of second-order elliptic systems in divergence form with rapidly oscillating almost-periodic coefficients, we obtain estimates for approximate correctors in terms of a function that quantifies the almost periodicity of the coefficients. The results are used to investigate the problem of convergence rates. We also establish uniform H\"older estimates for the Dirichlet problem in a bounded $C^{1, \alpha}$ domain.Comment: 41 pages; minor revision of the previous versio