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The Fractional Korn Inequality on Uniform Domains and New Korn Inequalities for Truncated Seminorms

Gabriel Acosta, Irene Drelichman, Ricardo Durán, Fernando López-García, Ignacio Ojea

TL;DR

This work addresses fractional Korn inequalities on non-smooth domains, proving the second case on uniform domains by first establishing a truncated-seminorm Korn inequality on John domains and then transferring the result to uniform domains through a norm equivalence between $W^{s,p}$ and its truncated counterpart. The core method combines a discrete Poincaré inequality on Whitney-derived trees with skew-symmetric matrix projections to control the full gradient by its symmetric part in a nonlocal, fractional setting. The authors also obtain weighted estimates with distance-to-boundary weights constrained by the fractional exponent and the Assouad codimension of the boundary, broadening applicability to domains with fractal boundaries. The results provide a robust framework for nonlocal elasticity models and fractional Sobolev regularity on irregular domains, and they introduce a versatile local-to-global strategy based on trees and smoothened cubes that may be useful beyond Korn-type problems.

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.

The Fractional Korn Inequality on Uniform Domains and New Korn Inequalities for Truncated Seminorms

TL;DR

This work addresses fractional Korn inequalities on non-smooth domains, proving the second case on uniform domains by first establishing a truncated-seminorm Korn inequality on John domains and then transferring the result to uniform domains through a norm equivalence between and its truncated counterpart. The core method combines a discrete Poincaré inequality on Whitney-derived trees with skew-symmetric matrix projections to control the full gradient by its symmetric part in a nonlocal, fractional setting. The authors also obtain weighted estimates with distance-to-boundary weights constrained by the fractional exponent and the Assouad codimension of the boundary, broadening applicability to domains with fractal boundaries. The results provide a robust framework for nonlocal elasticity models and fractional Sobolev regularity on irregular domains, and they introduce a versatile local-to-global strategy based on trees and smoothened cubes that may be useful beyond Korn-type problems.

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.
Paper Structure (6 sections, 12 theorems, 101 equations, 1 figure)

This paper contains 6 sections, 12 theorems, 101 equations, 1 figure.

Table of Contents

  1. Introduction
  2. A discrete Poincaré inequality on trees
  3. Domains and Trees
  4. Some preliminary results on skew-symmetric matrices
  5. Weighted truncated fractional Korn inequalities on John domains
  6. Fractional Korn inequality on uniform domains

Key Result

Lemma 2.1

Let $G=(V,E)$ be a rooted tree with root $t_0$ and ${\bm \nu}$, ${\bm \mu} \in \ell^1(V)$ two weights over $V$ such that the following Hardy-type inequality holds for every sequence ${\bm b}=\{b_t\}_{t\in V}$ over $V$: Then, the following weighted Poincaré-type inequality holds for all ${\bm b}$ indexed over $V$, where $\bar{b} = \frac{1}{\sum_{t\in V}\nu_t}\sum_{t\in V}b_t\nu_t$. The constant $

Figures (1)

  • Figure 1: A Whitney decomposition, with a possible underlying tree-structure. The shadow of the cube $Q_t$ is sketched.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Theorem 2.2
  • Remark 2.3
  • Definition 3.1: Uniform Domain (see for example Jones1981
  • Definition 3.2: John Domain (see for example ADbook
  • Lemma 3.3
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • ...and 17 more