Martingales, laminates and minimal Korn inequalities
Gabriele Cassese
TL;DR
The paper addresses how many scalar measurements are needed to bound the full gradient in Korn-type inequalities, reframing the problem via rank-one convexity and quasiconvexity. It develops a novel laminates–martingales connection to construct explicit witnesses, yielding sharp dimension bounds $N(d)=d^2-n_{\,\mathbb{R}}(d,1)$ with asymptotics $N(d)=2d(1-o(1))$ and $N'(d)=2d-1$, and provides a dimension-optimal Hankel-based Korn inequality. It also delivers a quantitative proof of Ornstein's non-inequality, extends the framework to rectangular and $L^p$ settings, and discusses the dependence of constants on dimension and domain. The approach blends calculus of variations, algebraic topology, and probabilistic martingale techniques to obtain sharp constants and broad applicability to Korn-type 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 $H_0^1(Ω)$ and $H^1(Ω)$, introducing the minimal numbers $N(d,Ω)$ and $N'(d,Ω)$ 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,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 $\frac{\nabla u+\nabla u^t}2$ and $\frac{\nabla u+\nabla u^t}2-\mathrm{div}(u)\frac{\mathrm{Id}}d$).