论文标题
$ l^p $ dirichlet和二阶椭圆系统的规律性问题，并适用于LaméSystem
The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system
论文作者
论文摘要
在论文Arxiv：1708.02289中，我们在Lipschitz图上方的域上引入了以差异形式强烈椭圆形的二阶系统的新可溶性方法，满足了$ l^p $ bugnary数据，$ l^p $ bugnary to $ p $接近$ 2 $。我们结果的主要新方面是，它适用于具有有限规律性系数的运算符，并适用于满足自然Carleson条件的操作员，该算病首先在标量案例中考虑过。 在本文中，我们将此结果扩展到多个方向。我们提高了$ l^p $ dirichlet问题的可解决性范围$ 2- \ varepsilon <p <p <\ frac {2（n-1）} {（n-3）}+\ varepsilon $，用于尺寸$ n = 2,3 $的系统，$ n = 2,3 $，范围为$ 2- \ 2- \ varepsilon <p <p <p <p <p <我们通过考虑规则性问题的可解决性（边界数据在$ l^p $中具有一个导数）来做到这一点。 其次，我们查看扰动类型 - 我们可以从已知的$ l^p $ dirichlet问题中推断出$ l^p $ dirichlet问题，从已知的$ l^p $ dirichlet dirichlet dirichlet dirichlet close close ocerator“ carleson Menate”（从carleson措施）中。在$ 2- \ varepsilon <p <\ frac {2（n-1）} {（n-2）}+\ varepsilon $的情况下，$ 2- \ varepsilon <p <\ frac {2（n-1）}+\ varepsilon $中的问题。 所有这些结果适用的系统的一个特定示例是poisson比率$ν<0.396 $的各向同性不均匀材料的Lamé操作员。在这种特定情况下，可以进一步改善可溶解性范围，请参阅J. Li和J. Pipher的即将进行的工作。
In the paper arXiv:1708.02289 we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domains above a Lipschitz graph, satisfying $L^p$-boundary data for $p$ near $2$. The main novel aspect of our result is that it applies to operators with coefficients of limited regularity and applies to operators satisfying a natural Carleson condition that has been first considered in the scalar case. In this paper we extend this result in several directions. We improve the range of solvability of the $L^p$ Dirichlet problem to the interval $2-\varepsilon < p<\frac{2(n-1)}{(n-3)}+\varepsilon$, for systems in dimension $n=2,3$ in the range $2-\varepsilon < p<\infty$. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in $L^p$) in the range $2-\varepsilon < p<2+\varepsilon$. Secondly, we look at perturbation type-results where we can deduce solvability of the $L^p$ Dirichlet problem for one operator from known $L^p$ Dirichlet solvability of a \lq\lq close" operator (in the sense of Carleson measure). This leads to improvement of the main result of the paper arXiv:1708.02289; we establish solvability of the $L^p$ Dirichlet problem in the interval $2-\varepsilon < p<\frac{2(n-1)}{(n-2)}+\varepsilon$ under a much weaker (oscillation-type) Carleson condition. A particular example of the system where all these results apply is the Lamé operator for isotropic inhomogeneous materials with Poisson ratio $ν<0.396$. In this specific case further improvements of the solvability range are possible, see the upcoming work with J. Li and J. Pipher.
