Almost-periodic homogenization of elliptic problems in non-smooth domains
Jun Geng, Bojing Shi · Proceedings of the American Mathematical Society · 2018
We consider a family of second-order elliptic operators { L ε } \{\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 W^{1,p} estimate of second-order elliptic systems holds for 2 n n + 1 − δ > p > 2 n n − 1 + δ \frac {2n}{n+1}-\delta >p>\frac {2n}{n-1}+\delta ; the ranges are sharp for n = 2 n=2 or n = 3 n=3 . In the scalar case we obtain that the W 1 , p W^{1,p} estimate holds for 3 2 − δ > p > 3 + δ \frac {3}{2}-\delta >p>3+\delta if n ⩾ 3 n\geqslant 3 , and 4 3 − δ > p >