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The Lp Dirichlet and regularity problems for second order elliptic systems with application to the Lamé system
Communications in Partial Differential Equations ( IF 1.9 ) Pub Date : 2021-03-01 , DOI: 10.1080/03605302.2021.1892131
Martin Dindoš 1
Affiliation  

Abstract

In a recent joint paper with Hwang and Mitrea we have introduced new solvability methods for strongly elliptic second order systems in divergence form on a domain above a Lipschitz graph, satisfying Lp-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 Lp Dirichlet problem to the interval 2ε<p<2(n1)(n3)+ε, for systems in dimension n = 2, 3 in the range 2ε<p<. We do this by considering solvability of the Regularity problem (with boundary data having one derivative in Lp) in the range 2ε<p<2+ε. Secondly, we look at perturbation type-results where we can deduce solvability of the Lp Dirichlet problem for one operator from known Lp Dirichlet solvability of a “close” operator (in the sense of Carleson measure). This leads to improvement of the main result of our previous paper; we establish solvability of the Lp Dirichlet problem in the interval 2ε<p<2(n1)(n2)+ε 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.



中文翻译:

二阶椭圆系统的 Lp Dirichlet 和正则问题在 Lamé 系统中的应用

摘要

在最近与 Hwang 和 Mitrea 的联合论文中,我们为 Lipschitz 图上方域上发散形式的强椭圆二阶系统引入了新的可解性方法,满足p接近 2 的L p边界数据。 我们结果的主要新颖方面是它适用于具有有限正则性系数的算子,并适用于满足在标量情况下首先考虑的自然卡尔森条件的算子。在本文中,我们将这个结果扩展到几个方向。我们将L p Dirichlet 问题的可解范围提高到区间2-ε<<2(n-1)(n-3)+ε,对于维度n  = 2, 3 在范围内的系统2-ε<<.我们通过考虑范围内的正则性问题(边界数据在L p 中具有一个导数)的可解性来做到这一点2-ε<<2+ε.其次,我们查看扰动类型结果,其中我们可以从“接近”算子的已知L p Dirichlet 可解性(在卡尔森测度的意义上)推导出一个算子的L p Dirichlet 问题的可解性。这导致了我们之前论文的主要结果的改进;我们建立了区间内L p Dirichlet 问题的可解性2-ε<<2(n-1)(n-2)+ε在更弱的(振荡型)卡尔森条件下。所有这些结果都适用的系统的一个特定示例是具有泊松比的各向同性非均匀材料的拉梅算子ν<0.396. 在这种特定情况下,可解范围的进一步改进是可能的。

更新日期:2021-03-01
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