Mathematics > Analysis of PDEs
[Submitted on 26 Apr 2022 (v1), last revised 8 Jan 2023 (this version, v3)]
Title:The Fractional Lamé-Navier Operator: Appearances, Properties and Applications
View PDFAbstract: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.
Submission history
From: James Scott [view email][v1] Tue, 26 Apr 2022 01:55:22 UTC (43 KB)
[v2] Fri, 11 Nov 2022 14:58:07 UTC (44 KB)
[v3] Sun, 8 Jan 2023 18:13:49 UTC (44 KB)
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