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Korn's inequality from the viewpoint of calculus of variations

Gabriele Cassese

Abstract

We study the best possible constants in Korn-type inequalities and their connection with Morrey's problem in the calculus of variations. We adapt techniques from the analysis of the Beurling-Ahlfors transform to Korn's inequality. In particular we show that the constant in Korn's inequality admits a dimension-free bound, and we obtain an estimate that is sharp up to a factor of $\sqrt 3$. In the radial case, the estimate is sharp. We also establish several improvements to estimates in various function spaces. Using a weighted version of Burkholder's differential subordination theorem, recently introduced in [J. Reine Angew. Math. 824 (2025), pp. 137-166],we also prove a dimension-free weighted version of the inequality for Muckenhoupt weights.

Korn's inequality from the viewpoint of calculus of variations

Abstract

We study the best possible constants in Korn-type inequalities and their connection with Morrey's problem in the calculus of variations. We adapt techniques from the analysis of the Beurling-Ahlfors transform to Korn's inequality. In particular we show that the constant in Korn's inequality admits a dimension-free bound, and we obtain an estimate that is sharp up to a factor of . In the radial case, the estimate is sharp. We also establish several improvements to estimates in various function spaces. Using a weighted version of Burkholder's differential subordination theorem, recently introduced in [J. Reine Angew. Math. 824 (2025), pp. 137-166],we also prove a dimension-free weighted version of the inequality for Muckenhoupt weights.
Paper Structure (17 sections, 33 theorems, 197 equations, 3 figures)

This paper contains 17 sections, 33 theorems, 197 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The setup and a first estimate
  4. The toolkit: Burkholder's method and calculus of variations
  5. A brief overview of Burkholder's theory
  6. The viewpoint of calculus of variations
  7. Proof of Theorem \ref{['thm:Korn']}
  8. Proof of the upper bound in Theorem \ref{['thm:Korn']}
  9. Lower bounds
  10. Particular cases of Korn's inequality: Full gradient, divergence-free and low dimensions
  11. Rank-one convexity, improved lower bounds and sharpness
  12. Korn in other function spaces
  13. Orlicz spaces
  14. Korn in weighted Sobolev spaces
  15. Concluding remarks
  16. ...and 2 more sections

Key Result

Theorem 1.1

Given $u\in \dot{W}^{1,p}(\mathbb{R}^d, \mathbb{R}^d)$ we have where $p^*=\max(p',p)$, $p'$ is the Hölder conjugate of $p$. In other words Moreover, The same result holds true if we consider the trace-free version of Korn's inequality for $d\ge 3$: Finally, if $u$ is generalised-radial, meaning that $u(x)=r(|x|)x$ for some $r\colon\mathbb{R}^+\to M_d(\mathbb{R})$, then we have the following st

Figures (3)

  • Figure 1: Schematic of the heat martingale $P_{T-t}f(\mathcal{B}_t)$
  • Figure 2: Upper and lower bounds for $\|R\otimes R\|_{L^p}$
  • Figure 3: The improvement over the natural bound

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Remark 1
  • Definition 1: Differentially subordinated martingales
  • Theorem 3.1: osekowski, Burkholder1988Sharp
  • ...and 61 more