[Paper Review] The Fractional Korn Inequality on Uniform Domains and New Korn Inequalities for Truncated Seminorms

Gabriel Acosta, Irene Drelichman+3 more|arXiv (Cornell University)|Jan 13, 2026

Analytic and geometric function theory0 citations

TL;DR

The paper proves the second case of the fractional Korn inequality on uniform domains and introduces a truncated seminorm framework that extends to John domains, with weighted estimates related to boundary distance.

ABSTRACT

We prove the so-called second case of the fractional Korn inequality for uniform domains. We obtain this result as an application of a novel fractional Korn-type inequality formulated in terms of truncated seminorms, which turns out to be valid for the broader class of John domains. We also obtain weighted estimates in which the weights are certain powers of the distance to the boundary that depend on the fractional exponent and the Assouad codimension of the boundary of the domain.

Motivation & Objective

• Motivate and formalize fractional Korn inequalities as nonlocal analogues of classical Korn inequalities.
• Establish the second case of the fractional Korn inequality on uniform domains.
• Develop a truncated seminorm approach that is valid on John domains and leads to weighted estimates.
• Connect local (cube) results to global domain classes via discrete Poincaré on trees.

Proposed method

• Formulate fractional Korn inequalities using Gagliardo and truncated X^{s,p} seminorms.
• Introduce truncated seminorms |u|_{W^{s,p}_{τ}} and |u|_{X^{s,p}_{τ}} to handle non-smooth domains.
• Use a discrete Poincaré inequality on trees to bridge local cube estimates to John/uniform domains.
• Analyze skew-symmetric projections onto RM and derive bounds relating |u-Π_Ω u|_{W^{s,p}} to |u|_{X^{s,p}}.
• Employ Whitney decompositions and smoothened cubes to transfer results from cubes to John/uniform domains.
• Develop weighted estimates with distance-to-boundary weights depending on s and Assouad codimension.

Experimental results

Research questions

• RQ1 Does the second (noncoercive) Korn inequality hold for uniform domains in the fractional (nonlocal) setting?
• RQ2 Can a truncated seminorm framework extend fractional Korn inequalities to the broader class of John domains?
• RQ3 What weighted formulations (distance to boundary) are valid in the fractional Korn context and how do Assouad codimension parameters come into play?
• RQ4 How can local cube estimates be extended to complex domains via tree-based discrete Poincaré inequalities?

Key findings

• The second case of the fractional Korn inequality holds on uniform domains.
• A truncated seminorm formulation yields a broader validity, extending results to John domains.
• Weighted estimates are obtained with weights as powers of the distance to the boundary, depending on s and Assouad codimension.
• A discrete Poincaré inequality on trees underpins the local-to-global argument, enabling extension from cubes to uniform/John domains.
• Preliminary results on skew-symmetric matrices and their interaction with smoothened cubes support the main inequalities.