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二阶复系数椭圆算子的边值问题

Boundary value problems for second order elliptic operators with complex coefficients

Martin Dindoš, Jill Pipher

备注：
41 pages. arXiv admin note: text overlap with arXiv:1612.01568
发表信息：
Analysis & PDE 13 (2020) 1897-1938
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摘要：

$\mathcal{L}=\mbox{div} A(x)\nabla$形式的二阶复系数算子理论最近在$p$ -椭圆性假设下得到了发展。特别地，如果矩阵$A$是$p$ -椭圆的，则解$u$至$\mathcal{L}u = 0$将满足更高的可积性，即使它们在内部可能不是连续的。此外，这些解决方案具有以下性质：$|u|^{p/2-1}u \in W^{1,2}_{loc}$。这些性质的解决方案被用于Dindoethan-Pipher解决$L^p$狄利克雷问题的$p$ -椭圆算子的系数满足进一步的正则性条件，一个Carleson措施的条件，经常出现在文献中的研究，真正的椭圆发散形式的运营商。本文包含两个主要结果。首先，我们建立了这类算子的正则性边值问题的可解性，在相同的范围内的狄利克雷问题。正则性问题，即使在真实的椭圆设置，是更微妙的比狄利克雷问题，因为它需要估计的导数的解决方案。其次，正则性的结果使我们能够扩展以前建立的狄利克雷问题的$L^p$可解性的范围，使用一个定理，由于Z。Shen关于一般有界次线性算子。

英文摘要：

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 Dindoš-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.

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