Download this article
 Download this article For screen
For printing
Recent Issues
Volume 12, Issue 1
Volume 11, Issue 4
Volume 11, Issue 3
Volume 11, Issue 2
Volume 11, Issue 1
Volume 10, Issue 4
Volume 10, Issue 3
Volume 10, Issue 2
Volume 10, Issue 1
Volume 9, Issue 4
Volume 9, Issue 3
Volume 9, Issue 2
Volume 9, Issue 1
Volume 8, Issue 4
Volume 8, Issue 3
Volume 8, Issue 2
Volume 8, Issue 1
Volume 7, Issue 4
Volume 7, Issue 3
Volume 7, Issue 2
Volume 7, Issue 1
Volume 6, Issue 4
Volume 6, Issue 3
Volume 6, Issue 2
Volume 6, Issue 1
Volume 5, Issue 3-4
Volume 5, Issue 2
Volume 5, Issue 1
Volume 4, Issue 3-4
Volume 4, Issue 2
Volume 4, Issue 1
Volume 3, Issue 4
Volume 3, Issue 3
Volume 3, Issue 2
Volume 3, Issue 1
Volume 2, Issue 2
Volume 2, Issue 1
Volume 1, Issue 2
Volume 1, Issue 1
The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 2325-3444 (e-only)
ISSN: 2326-7186 (print)
Author Index
To Appear
 
Other MSP Journals
Fractional Korn's inequalities without boundary conditions

Davit Harutyunyan, Tadele Mengesha, Hayk Mikayelyan and James M. Scott

Vol. 11 (2023), No. 4, 497–524
Abstract

Motivated by a linear nonlocal model of elasticity, this work establishes fractional analogues of Korn’s first and second inequalities for vector fields in fractional Sobolev spaces defined over a bounded domain. The validity of the inequalities require no additional boundary condition, extending existing fractional Korn’s inequalities that are only applicable for Sobolev vector fields satisfying zero Dirichlet boundary conditions. The domain of definition is required to have a C1-boundary or, more generally, a Lipschitz boundary with small Lipschitz constant. We conjecture that the inequalities remain valid for vector fields defined over any Lipschitz domain. We support this claim by presenting a proof of the inequalities for vector fields defined over planar convex domains.

Keywords
fractional Korn inequality, fractional Hardy inequality, peridynamics
Mathematical Subject Classification
Primary: 45F99, 74B99, 74G65
Milestones
Received: 28 February 2023
Revised: 5 September 2023
Accepted: 3 October 2023
Published: 1 December 2023

Communicated by Micol Amar
Authors
Davit Harutyunyan
Department of Mathematics
University of California Santa Barbara
Santa Barbara, CA
United States
Tadele Mengesha
Department of Mathematics
University of Tennessee Knoxville
Knoxville, TN
United States
Hayk Mikayelyan
School of Mathematical Sciences
University of Nottingham Ningbo
Ningbo
China
James M. Scott
Department of Applied Physics and Applied Mathematics
Columbia University
New York, NY
United States