We consider a family of second-order elliptic operators {L ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in ℝn. We are able to show that the uniform W 1,p estimate of second order elliptic systems holds for $$\frac{{2n}}{{n + 1}} - \delta < p < \frac{{2n}}{{n - 1}} + \delta $$ 2 n n + 1 − δ < p < 2 n n − 1 + δ where δ > 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W 1,p estimate is true for $$\frac{3}{2} - \delta < p < 3 + \delta $$ 3 2 − δ < p < 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 < p < ∞.
Acta Mathematica Sinica, English Series – Springer Journals
Published: Nov 28, 2017
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