For every p>1 there are p-homogeneous rank-one convex integrands on R^{2xm}, and on large square matrix spaces, that are nowhere quasiconvex; for m large enough and in R^{4x2} when p≠2.
Korn's inequality from the viewpoint of calculus of variations
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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 obtain a bound that is sharp up to a factor of $1.158$. 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.
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A solution to Morrey's problem in $\mathbb{R}^{2\times m}$
For every p>1 there are p-homogeneous rank-one convex integrands on R^{2xm}, and on large square matrix spaces, that are nowhere quasiconvex; for m large enough and in R^{4x2} when p≠2.