Korn's inequality from the viewpoint of calculus of variations

2026/03/31 by Gabriele Cassese
#math.AP

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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 dimension 2, we find the sharp L^p constant: p^*-1. In general, 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 √ 3. We also establish several improvements to estimates in various other 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.

Citations

• Martingales, laminates and minimal Korn inequalities
• Lower semicontinuity, Stoilow factorization and principal maps
• The local Burkholder functional, quasiconvexity and Geometric Function Theory
• Korn and Poincaré-Korn inequalities: A different perspective
• Continuous sparse domination and dimensionless weighted estimates for the Bakry Riesz vector
• Optimal range of Haar martingale transforms and its applications
• Weighted square function estimates
• Trace-free Korn inequalities in Orlicz spaces
• Korn's inequality and John domains
• On the optimal constants in Korn's and geometric rigidity estimates, in bounded and unbounded domains, under Neumann boundary conditions
• Martingales and Sharp Bounds for Fourier multipliers
• Sharp Lower bound estimates for vector-valued and matrix-valued multipliers in L^p
• Martingale transforms and their projection operators on manifolds
• Laminates Meet Burkholder Functions
• The foundational inequalities of D.L. Burkholder and some of their\n ramifications
• The sharp weighted bound for general Calderon-Zygmund operators
• Burkholder's function via Monge--Ampère equation
• On the norm of the Beurling-Ahlfors operator in several dimensions
• On singular integral and martingale transforms
• Heating of the Ahlfors-Beurling operator: weakly quasiregular maps on the plane are quasiregular

Cited by

• Martingales, laminates and minimal Korn inequalities

Related

• A note on the Korn inequality in a n-dimensional context
• Martingales, laminates and minimal Korn inequalities
• Korn and Poincaré-Korn inequalities: A different perspective
• Weighted asymptotic Korn and interpolation Korn inequalities with singular weights
• Korn inequality on irregular domains