Consistency in fractional Korn's inequalities on Lipschitz domains
Abstract
We investigate fractional analogs of Korn's inequality on bounded Lipschitz domains in several contexts. Constrained and unconstrained inequalities are established; constraints include quotient spaces and spaces incorporating various classes of boundary conditions or volume constraints. We also expand the parameters of integrability for which the inequalities hold. Specifically, a fractional Korn's first unconstrained inequality holds for vector fields in the fractional Sobolev space Ws,p(Ω;ℝn) for all p ∈ [1,∞] and all s ∈ (0,1). In the cases p = 1 and p = ∞, the inequality has no local analog. We quantify this by investigating the consistency of the inequalities as the differentiability parameter s → 1; the universal constant in the inequality remains stable for p ∈ (1,∞) but degenerates for p ∈ {1,∞}. We introduce an explicit Lp-bounded projection operator onto infinitesimal rigid motions as a key tool to quantitatively strengthen the inequality.
Related
- Korn and Poincaré-Korn inequalities for thin domains in GSBDp
- Boundary Harnack inequalities for Kolmogorov equations in asymptotically cylindrical Lipschitz domains
- Optimal Global Sobolev Estimates for Fractional Dirichlet Problems in Bounded Lipschitz Domains
- Boundary Harnack inequalities for the fractional p-Laplacian in Reifenberg flat domains
- Weighted heat traces of the Dirichlet Laplacian on Lipschitz domains
BibTeX
@misc{mengesha2026consistency,
title = {{Consistency in fractional Korn's inequalities on Lipschitz domains}},
author = {Tadele Mengesha and James M. Scott},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.33325},
eprint = {2610.33325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.33325},
}
RIS
TY - UNPB
TI - Consistency in fractional Korn's inequalities on Lipschitz domains
AU - Tadele Mengesha
AU - James M. Scott
PY - 2026
DA - 2026/10/23
PB - arXiv
DO - 10.48550/arxiv.2610.33325
UR - https://arxiv.org/abs/2610.33325
AB - We investigate fractional analogs of Korn's inequality on bounded Lipschitz domains in several contexts. Constrained and unconstrained inequalities are established; constraints include quotient spaces and spaces incorporating various classes of boundary conditions or volume constraints. We also expand the parameters of integrability for which the inequalities hold. Specifically, a fractional Korn's first unconstrained inequality holds for vector fields in the fractional Sobolev space W^s,p(Ω;ℝ^n) for all p ∈ [1,∞] and all s ∈ (0,1). In the cases p = 1 and p = ∞, the inequality has no local analog. We quantify this by investigating the consistency of the inequalities as the differentiability parameter s → 1; the universal constant in the inequality remains stable for p ∈ (1,∞) but degenerates for p ∈ {1,∞}. We introduce an explicit L^p-bounded projection operator onto infinitesimal rigid motions as a key tool to quantitatively strengthen the inequality.
ER -