Almost-periodic homogenization of elliptic problems in non-smooth domains

Abstract:

We consider a family of second-order elliptic operators $\{\mathcal {L}_\varepsilon \}$ in divergence form with rapidly oscillating and almost-periodic coefficients in Lipschitz domains. By using the compactness method, we show that the uniform $W^{1,p}$ estimate of second-order elliptic systems holds for $\frac {2n}{n+1}-\delta <p<\frac {2n}{n-1}+\delta$; the ranges are sharp for $n=2$ or $n=3$. In the scalar case we obtain that the $W^{1,p}$ estimate holds for $\frac {3}{2}-\delta <p<3+\delta$ if $n\geqslant 3$, and $\frac {4}{3}-\delta <p<4+\delta$ if $n=2$; the ranges of $p$ are sharp.