论文标题
分数Lamé-Navier操作员：出现，属性和应用程序
The Fractional Lamé-Navier Operator: Appearances, Properties and Applications
论文作者
论文摘要
我们介绍并分析了部分差分运算符Lamé-Navier系统的明确表述。我们表明，该分数Lamé-navier操作员是一个非局部间差异操作员，它出现在几种广泛使用的连续性力学模型中。我们证明可以使用非局部梯度算子的组合物获得分数Lamé-navier操作员。此外，分数Lamé-navier操作员的有效形式与在基于状态的Peridynamics中获得的特定参数选择的操作员相同。我们进一步表明，与局部经典拉梅 - 纳维尔系统相关的dirichlet to-neumann地图与半空间相吻合，与Lamé-navier操作员的平方根功率相吻合，以特别选择弹性系数。 我们为分数Lamé-Navier操作员建立了基本分析结果，包括阳性和负功率的计算，并探索其与Hölder和Bessel功能类别的相互作用。我们还将分数拉梅 - 纳维尔（Lamé-navier）推导为上半空间中局部退化椭圆形的方程式的迪利奇特（Dirichlet）图。我们为扩展系统的泊松内核使用明确的公式来建立加权Sobolev空间中的良好性。作为应用程序，我们使用扩展系统中的纯局部参数得出了两个分数sInorms的等效性，然后使用此等效性获得与分数Lamé-Navier操作员相关的变异性Dirichlet问题的良好性。
We introduce and analyze an explicit formulation of fractional powers of the Lamé-Navier system of partial differential operators. We show that this fractional Lamé-Navier operator is a nonlocal integro-differential operator that appears in several widely-used continuum mechanics models. We demonstrate that the fractional Lamé-Navier operator can be obtained using compositions of nonlocal gradient operators. Additionally, the effective form of the fractional Lamé-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 Lamé-Navier system posed in a half-space coincides with the square root power of the Lamé-Navier operator for a particular choice of elastic coefficients. We establish basic analysis results for the fractional Lamé-Navier operator, including the calculus of positive and negative powers, and explore its interaction with the Hölder and Bessel classes of functions. We also derive the fractional Lamé-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 Lamé-Navier operator.
