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 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 $\sqrt 3$.

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.