Korn's inequality from the viewpoint of calculus of variations

1. Problem Statement

The paper addresses the problem of determining the optimal constants in Korn's inequalities, which relate the $L^p$ norm of the skew-symmetric part of the gradient of a vector field to the $L^p$ norm of its symmetric part (the strain tensor).

Specifically, for a domain $\Omega \subset \mathbb{R}^d$ and $u \in W^{1,p}_0(\Omega, \mathbb{R}^d)$, the inequality is:
$$\|A(u)\|_{L^p} \leq C(p, \Omega) \|E(u)\|_{L^p}$$
where $A(u) = \frac{1}{2}(\nabla u - \nabla u^T)$ is the skew-symmetric part and $E(u) = \frac{1}{2}(\nabla u + \nabla u^T)$ is the symmetric part.

Key Challenges:

• Dimension Dependence: Previous bounds for the constant $C(p, d)$ often depended on the dimension $d$. The paper seeks a dimension-free bound.
• Sharpness: While the lower bound $C(p, d) \geq p^* - 1$ (where $p^* = \max(p, p')$) was known, the upper bound was not sharp. The best known upper bounds were either dimension-dependent or had large multiplicative constants.
• Connection to Variational Problems: The optimal constant is linked to the quasiconvexity of certain integrands, a central open problem in the calculus of variations known as Morrey's problem.

2. Methodology

The author employs a novel synthesis of stochastic analysis (martingale theory) and the calculus of variations.

A. Singular Integrals and Martingale Representation

The paper establishes that Korn's inequality is equivalent to bounding the operator norm of the tensor product of Riesz transforms, $R \otimes R$.

• The operator $R \otimes R$ is represented using a heat martingale approach. By defining a space-time Brownian motion $Y^T_t = (B_t, T-t)$ and the heat extension $Uf$, the Riesz transforms are recovered as limits of conditional expectations of martingales.
• This allows the application of Burkholder's differential subordination theory. If a martingale $X$ is differentially subordinated to $Y$ (meaning $|dX| \leq |dY|$), then $\|X\|_p \leq (p^*-1)\|Y\|_p$.

B. Tensorialization and Orthogonality

A major technical hurdle is extending scalar martingale bounds to matrix-valued functions without introducing dimension-dependent factors.

• The author constructs a specific matrix-valued martingale $Z^T_t$ associated with the gradient.
• By leveraging the orthogonality of Brownian motion coordinates, the author proves that the quadratic variation of the skew-symmetric martingale is bounded by the quadratic variation of the symmetric martingale, up to a constant factor derived from linear algebra.

C. Calculus of Variations and Rank-One Convexity

The paper bridges the gap between the martingale bounds and the sharpness of the constants:

• It relates the constant $C(p, d)$ to the quasiconvexity of the integrand $f(A) = C|PSym(A)|^p - |PSkew(A)|^p$.
• It utilizes the connection between rank-one convexity and dyadic martingales with rank-one increments.
• The author proves that the rank-one convex envelope of the integrand is exactly determined by Burkholder's function $G_p$, which is the optimal function for the differential subordination inequality.

D. Weighted Estimates

The methodology is extended to Muckenhoupt $A_p$ weights by utilizing a recent weighted version of Burkholder's theorem (Domelevo, Petermichl, and Skreb, 2025) and defining a "heat weight" class $A^{Heat}_p$ equivalent to the standard $A_p$ class.

3. Key Contributions and Results

A. Dimension-Free Upper Bounds

The paper proves the following upper bound for the optimal constant $C(p, d)$:
$$C(p, d) \leq \sqrt{3}(p^* - 1)$$

• This bound is independent of the dimension $d$.
• It improves upon previous results (e.g., Hytönen's $(d+1)(p^*-1)/2$) by removing the linear dependence on $d$.
• The factor $\sqrt{3}$ is shown to be sharp for the general case, though the author conjectures the true sharp constant is $p^* - 1$.

B. Trace-Free and Full Gradient Versions

• Trace-free version: For $d \geq 3$, the same bound holds: $C_0(p, d) \leq \sqrt{3}(p^* - 1)$.
• Full gradient version: The paper provides improved bounds for the constant $C_{KN}(p, d)$ relating the full gradient $\nabla u$ to the strain $E(u)$.

C. Radial Case Sharpness

For generalized-radial functions ($u(x) = r(|x|)x$), the bound is significantly improved:
$$\|A(u)\|_{L^p} \leq (p^* - 1) \|E(u)\|_{L^p}$$

• In the case $d=2$ and $p \geq 2$, this constant $(p^* - 1)$ is sharp (cannot be improved).
• This result supports the conjecture that $p^* - 1$ is the sharp constant for all functions, not just radial ones.

D. Lower Bounds and Rank-One Convexity

• The paper proves that the rank-one convex constant $C_{rc}(p, d)$ is exactly $p^* - 1$.
• This is achieved by constructing specific martingale sequences (witnesses) that demonstrate the lower bound.
• The result implies that the obstruction to the sharp constant being $p^* - 1$ lies entirely in the difference between rank-one convexity and quasiconvexity (Morrey's problem).

E. Orlicz and Weighted Spaces

• Orlicz Spaces: The results are extended to Sobolev-Orlicz spaces $W^{1, L^\Phi}$, providing dimension-free bounds dependent on the Simonenko indices of the Young function $\Phi$.
• Weighted Spaces: A dimension-free weighted inequality is established for $A_p$ weights:
  $$\|A(u)\|_{L^p(w)} \leq K(p^* - 1)[w]_{A_p}^{\max(1, \frac{1}{p-1})} \|E(u)\|_{L^p(w)}$$

4. Significance and Implications

1. Resolution of Dimensional Dependence: The paper successfully removes the linear dependence on dimension $d$ from the upper bounds of Korn's inequality, a significant step forward in harmonic analysis and elasticity theory.
2. Connection to Morrey's Problem: It provides a clear roadmap for the Iwaniec-Martin conjecture (that the sharp constant is $p^*-1$). The paper demonstrates that proving the sharpness of Korn's inequality is equivalent to proving that a specific rank-one convex integrand is quasiconvex.
3. New Tools for Variational Calculus: By successfully applying Burkholder's martingale methods to the tensor-valued setting of Korn's inequality, the paper offers a new toolkit for analyzing quasiconvexity and rank-one convexity in higher dimensions.
4. Applications: The improved constants lead to stronger coercivity estimates in linearized elasticity and fluid mechanics, potentially yielding new existence and regularity results for variational problems where $p^*-1$ is large.

Conclusion

Gabriele Cassese's paper represents a major advancement in the quantitative theory of Korn's inequalities. By adapting techniques from the analysis of the Beurling-Ahlfors transform and utilizing the deep connection between martingale theory and the calculus of variations, the author establishes a dimension-free upper bound of $\sqrt{3}(p^*-1)$ and proves that the rank-one convex constant is exactly $p^*-1$. This work narrows the gap between known upper and lower bounds and clarifies the role of quasiconvexity in determining the sharp constants of fundamental inequalities in elasticity.