Korn's inequality from the viewpoint of calculus of variations

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.