The theory of second order complex coefficient operators of the form $\mathcal{L}=\mbox{div} A(x)\nabla$ has recently been developed under the assumption of $p$-ellipticity. In particular, if the matrix $A$ is $p$-elliptic, the solutions $u$ to $\mathcal{L}u = 0$ will satisfy a higher integrability, even though they may not be continuous in the interior. Moreover, these solutions have the property that $|u|^{p/2-1}u \in W^{1,2}_{loc}$. These properties of solutions were used by Dindo\v{s}-Pipher to solve the $L^p$ Dirichlet problem for $p$-elliptic operators whose coefficients satisfy a further regularity condition, a Carleson measure condition that has often appeared in the literature in the study of real, elliptic divergence form operators. This paper contains two main results. First, we establish solvability of the Regularity boundary value problem for this class of operators, in the same range as that of the Dirichlet problem. The Regularity problem, even in the real elliptic setting, is more delicate than the Dirichlet problem because it requires estimates on derivatives of solutions. Second, the Regularity results allow us to extend the previously established range of $L^p$ solvability of the Dirichlet problem using a theorem due to Z. Shen for general bounded sublinear operators.

中文翻译：

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复系数二阶椭圆算子的边值问题

形式 $\mathcal{L}=\mbox{div} A(x)\nabla$ 的二阶复系数算子的理论最近在 $p$-椭圆率的假设下得到发展。特别是，如果矩阵 $A$ 是 $p$-椭圆，$\mathcal{L}u = 0$ 的解 $u$ 将满足更高的可积性，即使它们在内部可能不连续。此外，这些解决方案具有 $|u|^{p/2-1}u \in W^{1,2}_{loc}$ 的性质。Dindo\v{s}-Pipher 使用解的这些性质来解决 $p$-椭圆算子的 $L^p$Dirichlet 问题，其系数满足进一步的正则条件，这是经常出现在研究实椭圆散度形式算子的文献。本文包含两个主要结果。第一的，我们建立了此类算子的正则边值问题的可解性，与狄利克雷问题的范围相同。正则性问题，即使在真正的椭圆环境中，也比狄利克雷问题更为微妙，因为它需要对解的导数进行估计。其次，正则性结果允许我们使用 Z. Shen 的定理扩展先前建立的 Dirichlet 问题的 $L^p$ 可解性范围，用于一般有界次线性算子。