Martingales, laminates and minimal Korn inequalities

Korn’s inequalities show that the $L^2$-norm of $\mathrm{\nabla }u$ can be controlled by the $L^2$-norm of $\mathrm{S}\mathrm{y}\mathrm{m}(\mathrm{\nabla }u)$, which only has $d(d+1)\mathrm{/}2$ components. In [J. Math. Pures Appl. 148 (2021), pp. 199-220] Chipot posed the question of $\mathit{\text{how}}\mathit{\text{many}}\mathit{\text{scalar}}\mathit{\text{measurements}}\mathit{\text{are}}\mathit{\text{needed}}\mathit{\text{to}}\mathit{\text{have}}\mathit{\text{a}}\mathit{\text{Korn-type}}\mathit{\text{control}}\mathit{\text{on}}\mathrm{\nabla }u$ when $u$ is in $H_0^1(\mathrm{\Omega })$ and $H^1(\mathrm{\Omega })$, introducing the minimal numbers $N(d,\mathrm{\Omega })$ and $N^{\mathrm{\prime }}(d,\mathrm{\Omega })$ respectively. He proved general bounds and calculated several low-dimensional values of $N,N^{\mathrm{\prime }}$. We reframe Chipot’s problem in the language of rank-one convexity and quasiconvexity and obtain a purely algebraic characterisation of when such inequalities hold, which yields the sharp bounds \begin{align*} N(d,Ω)&=2d(1-o(1))\ N’(d,Ω)&=2d-1. \end{align*} As a consequence, we recover and streamline several of Chipot’s results, we obtain a dimension-optimal Korn inequality and several sharp estimates for the best constant for various Korn-type inequalities. Generalisations to the rectangular case and to general $L^p$ estimates are also considered.\par The central new ingredient of our approach is a systematic connection between laminates and martingales which produces explicit families of laminates realising these bounds. This method is of independent interest in the calculus of variations: for instance, we use it to obtain a new quick and quantitative proof of Ornstein’s non-inequality, valid for all first order homogeneous operators in ${\mathbb{R}}^{2\times 2}$ and for a large class of operators in general dimensions (including Korn’s $\frac{\mathrm{\nabla }u+\mathrm{\nabla }{u}^t}{2}$ and $\frac{\mathrm{\nabla }u+\mathrm{\nabla }{u}^t}{2}-\mathrm{d}\mathrm{i}\mathrm{v}(u)\frac{\mathrm{I}\mathrm{d}}{d}$).