A fractional Korn-type inequality

2018/08/06 by James M. Scott, Scott, James, Tadele Mengesha +1 · 2 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in engineering

paper · pdf · doi:10.48550/arxiv.1808.02133

openalex publication_date 2018/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that a class of spaces of vector fields whose semi-norms involve the magnitude of "directional" difference quotients is in fact equivalent to the class of fractional Sobolev spaces. The equivalence can be considered a Korn-type characterization of fractional Sobolev spaces. We use the result to understand better the energy space associated to a strongly coupled system of nonlocal equations related to a nonlocal continuum model via peridynamics. Moreover, the equivalence permits us to apply classical space embeddings in proving that weak solutions to the nonlocal system enjoy both improved differentiability and improved integrability.

Citations

• Reformulation of elasticity theory for discontinuities and long-range forces
• Peridynamic States and Constitutive Modeling
• On the variational limit of a class of nonlocal functionals related to peridynamics

Cited by

• A note on the L^p-solvability of a strongly-coupled nonlocal system of equations
• Homogenization of rate-independent elastoplastic spring network models with non-local random fields

Related

• A Fractional Korn-type inequality for smooth domains and a regularity estimate for nonlinear nonlocal systems of equations
• On the fractional Korn inequality in bounded domains: Counterexamples to the case ps<1
• Korn's inequality in anisotropic Sobolev spaces
• Homothetic variant of fractional Sobolev space with application to Navier-Stokes system revisited
• Nonlocal gradients in bounded domains motivated by Continuum Mechanics: Fundamental Theorem of Calculus and embeddings