Given a domain above a Lipschitz graph, we establish solvability results for strongly elliptic second-order systems in divergence-form, allowed to have lower-order (drift) terms, with $L^p$-boundary data for $p$ near $2$ (more precisely, in an interval of the form $\big(2-\varepsilon,\frac{2(n-1)}{n-2}+\varepsilon\big)$ for some small $\varepsilon>0$). The main novel aspect of our result is that the coefficients of the operator do not have to be constant, or have very high regularity, instead they will satisfy a natural Carleson condition that has appeared first in the scalar case. A significant example of a system to which our result may be applied is the Lam\'e operator for isotropic inhomogeneous materials. Dealing with genuine systems gives rise to substantial new challenges, absent in the scalar case. Among other things, there is no maximum principle for general elliptic systems, and the De Giorgi - Nash - Moser theory may also not apply. We are, nonetheless, successful in establishing estimates for the square-function and the nontangential maximal operator for the solutions of the elliptic system described earlier, and use these as alternative tools for proving $L^p$ solvability results for $p$ near $2$.

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具有粗糙系数的二阶椭圆系统的 $L^p$ Dirichlet 边界问题

给定 Lipschitz 图上方的域，我们为散度形式的强椭圆二阶系统建立可解性结果，允许具有低阶（漂移）项，$L^p$-边界数据为 $p$ 接近 $2 $（更准确地说，在 $\big(2-\varepsilon,\frac{2(n-1)}{n-2}+\varepsilon\big)$ 形式的区间中，对于一些小 $\varepsilon>0 $）。我们结果的主要新颖之处在于算子的系数不必是常数，也不必具有非常高的规律性，相反，它们将满足首先出现在标量情况下的自然卡尔森条件。可以应用我们的结果的系统的一个重要示例是各向同性非均匀材料的 Lam\'e 算子。处理真正的系统会带来大量的新挑战，而在标量情况下则没有。除其他事项外，一般椭圆系统没有极大值原理，De Giorgi-Nash-Moser 理论也可能不适用。尽管如此，我们还是成功地为前面描述的椭圆系统的解的平方函数和非切向极大值算子建立了估计，并使用这些作为替代工具来证明 $p$ 接近 $2 的 $L^p$ 可解性结果$.