Martingales, laminates and minimal Korn inequalities

Abstract

Korn's inequalities show that the $L^2$-norm of $\nabla u$ can be controlled by the $L^2$-norm of $\mathrm{Sym}(\nabla u)$, which only has $d(d+1)/2$ components. In [J. Math. Pures Appl. 148 (2021), pp. 199-220] Chipot posed the question of \textit{how many scalar measurements are needed to have a Korn-type control on $\nabla u$} when $u$ is in $H_0^1(\Omega)$ and $H^1(\Omega)$, introducing the minimal numbers $N(d,\Omega)$ and $N'(d,\Omega)$ respectively. He proved general bounds and calculated several low-dimensional values of $N,N'$. 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,\Omega)&=2d(1-o(1))\\ N'(d,\Omega)&=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 $\frac{\nabla u+\nabla u^t}2$ and $\frac{\nabla u+\nabla u^t}2-\mathrm{div}(u)\frac{\mathrm{Id}}d$).