Applications of Fourier analysis in homogenization of Dirichlet problem II. $L^p$ estimates
Hayk Aleksanyan, Per S Sjölin, Henrik Shahgholian · arXiv (Cornell University) · 2012
Let $u_\e$ be a solution to the system $$ \mathrm{div}(A_\e(x) abla u_{\e}(x))=0 \text{\ in} D, \qquad u_{\e}(x)=g(x,x/\e) \text{\ on}\partial D, $$ where $D \subset \R^d $ ($d \geq 2$), is a smooth uniformly convex domain, and $g$ is 1-periodic in its second variable, and both $A_\e$ and $g$ reasonably smooth. Our results in this paper are two folds. First we prove $L^p$ convergence results for solutions of the above system, for non-oscillating operator, $A_\e(x) =A(x)$, with the following convergence rate for all $1\leq p