The Fractional Lamé-Navier Operator: Appearances, Properties and Applications
The Fractional Lamé-Navier Operator: Appearances, Properties and Applications
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分数 Lamé-Navier 算子：外观、性质和应用
DOI：
10.2139/ssrn.4384323
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
J. Scott
中科院分区：
文献类型：
--
作者：
J. Scott
我们引入并分析了偏微分算子的LAM-E-Navier系统的分数次幂的一个显式公式。我们证明了这个分数阶Lam‘e-Navier算子是一种非局域积分-微分算子，它出现在几个广泛使用的连续介质力学模型中。我们证明了分数阶LAM‘e-Navier算子可以由非局部梯度算子的合成得到。此外，分数阶Lam‘e-Navier算符的有效形式与在基于状态的动力学中作为特定参数选择而得到的算符相同。我们进一步证明了在半空间中构造的局部经典Lam‘e-Navier系统的Dirichlet-to-Neumann映射与LAM’e-Navier算符对于特定弹性系数的平方幂重合。我们建立了分数阶Lam‘e-Navier算子的基本分析结果，包括正、负幂演算，并研究了它与H“old和Bessel函数类之间的相互作用。我们还得到了分数阶LAM‘e-Navier作为上半空间中局部退化椭圆型方程组的Dirichlet-to-Neumann映射。我们利用扩张系统的Poisson核的显式公式，建立了加权Sobolev空间中的适定性。作为应用，我们利用扩张系统中的一个纯局部变元得到了两个分式半范的等价性，然后利用这个等价性得到了与分式LAM‘e-Navier算子相关的变分Dirichlet问题的适定性。
We introduce and analyze an explicit formulation of fractional powers of the Lam\'e-Navier system of partial differential operators. We show that this fractional Lam\'e-Navier operator is a nonlocal integro-differential operator that appears in several widely-used continuum mechanics models. We demonstrate that the fractional Lam\'e-Navier operator can be obtained using compositions of nonlocal gradient operators. Additionally, the effective form of the fractional Lam\'e-Navier operator is the same as the operator obtained as a particular choice of parameters in state-based peridynamics. We further show that the Dirichlet-to-Neumann map associated to the local classical Lam\'e-Navier system posed in a half-space coincides with the square root power of the Lam\'e-Navier operator for a particular choice of elastic coefficients. We establish basic analysis results for the fractional Lam\'e-Navier operator, including the calculus of positive and negative powers, and explore its interaction with the H\"older and Bessel classes of functions. We also derive the fractional Lam\'e-Navier as the Dirichlet-to-Neumann map of a local degenerate elliptic system of equations in the upper half-space. We use an explicit formula for a Poisson kernel for the extension system to establish the well-posedness in weighted Sobolev spaces. As an application, we derive the equivalence of two fractional seminorms using a purely local argument in the extension system, and then use this equivalence to obtain well-posedness for a variational Dirichlet problem associated to the fractional Lam\'e-Navier operator.
影响因子：
1.7
作者：
Yuan Gao;Jian‐Guo Liu;Zibu Liu
通讯作者：
Yuan Gao;Jian‐Guo Liu;Zibu Liu
重新审视希尔伯特空间中刃位错的 Peierls-Nabarro 模型
DOI：
10.3934/dcdsb.2020224
发表时间：
2021
期刊：
Discrete & Continuous Dynamical Systems - B
影响因子：
--
作者：
Gao, Yuan;Liu, Jian-Guo;Luo, Tao;Xiang, Yang
通讯作者：
Xiang, Yang
影响因子：
1.3
作者：
Mengesha, Tadele;Scott, James M.
通讯作者：
Scott, James M.