Geometric lower bounds on Grad's number in spatial dimensions four and higher

Grad's number $G(\Omega)$, introduced by Desvillettes and Villani in their study of a Korn-type inequality with tangency boundary condition arising in the trend to equilibrium for the Boltzmann equation, quantifies how far a bounded domain $\Omega\subset\mathbb{R}^N$ is from being axisymmetric: it is the infimum of $\|\nabla^{\mathrm{sym}}v\|_{L^2(\Omega)}^2$ over divergence-free vector fields $v$ with prescribed antisymmetric gradient $\Sigma$ and $v\cdot n=0$ on $\partial\Omega$, minimized over all unit-norm antisymmetric matrices $\Sigma$. Positivity of $G(\Omega)$ characterizes non-axisymmetry in every dimension $N\ge 2$, and quantitative lower bounds on $G(\Omega)$ control the constant in the Korn-type inequality, hence the convergence rates to equilibrium. Explicit geometric lower bounds — Figalli's combination of the trace constant of $\Omega$ with the oscillation of $|x|^2$ on boundary components ($N=2$) or on plane slices of the boundary ($N=3$) — are known only in dimensions two and three. Figalli noted that his techniques might extend to $N\ge 4$; the extension has never been carried out. The problem is to establish comparable explicit geometric lower bounds on Grad's number for non-axisymmetric domains in dimensions $N\ge 4$.

Background

The Boltzmann equation governs the phase-space density of a dilute gas. When the gas is confined in a bounded container $\Omega\subset\mathbb{R}^N$ whose walls reflect molecules specularly (the incoming velocity is the elastic reflection of the outgoing one, $v\mapsto v-2(v\cdot n(x))\,n(x)$, where $n(x)$ is the outer unit normal at $x\in\partial\Omega$), the large-time behavior of solutions is strongly constrained by the shape of $\Omega$. In a generic container the only steady states are rest states (constant density and temperature, no macroscopic velocity), but if $\Omega$ has an axis of symmetry, rotating equilibria exist because the corresponding component of angular momentum is an additional conserved quantity. Desvillettes and Villani, in their proof of convergence to equilibrium with faster-than-any-polynomial rate for the Boltzmann equation, therefore had to quantify how far the container is from being axisymmetric.

The analytic device is a Korn-type inequality. Classical Korn's inequality bounds the full gradient of a vector field by its symmetric part, $\nabla u=\nabla^{\mathrm{sym}}u+\nabla^{a}u$ with $(\nabla^{\mathrm{sym}}u)_{ij}=\frac{1}{2}(\partial_j u_i+\partial_i u_j)$ and $\nabla^a u$ the antisymmetric part, under a no-slip (Dirichlet) boundary condition. Desvillettes and Villani proved a variant in which the boundary condition is pure tangency, $u\cdot n=0$ on $\partial\Omega$, the natural condition for displacement fields generated by specular transport: for every bounded $C^1$ domain $\Omega\subset\mathbb{R}^N$, $N\ge 2$, that is not axisymmetric, there exists $K(\Omega)>0$ such that

$\|\nabla^{\mathrm{sym}}u\|_{L^2(\Omega)}^2\ \ge\ K(\Omega)\,\|\nabla u\|_{L^2(\Omega)}^2$

for all tangent fields $u$ with $\nabla u\in L^2(\Omega)$. Here "axisymmetric" is understood in the general sense of Desvillettes–Villani (Definition–Lemma 1): $\Omega$ is invariant under a nontrivial one-parameter group $(e^{tR})_{t\in\mathbb{R}}$ of isometries whose linear part is antisymmetric (in $N=2$ this means circular symmetry about a point; in $N=3$, symmetry about an axis). The constant $K(\Omega)$ degenerates exactly as $\Omega$ approaches such a symmetric shape, and its size enters the convergence rates, so controlling $K(\Omega)$ through the geometry of $\Omega$ is the core quantitative problem.

Desvillettes and Villani reduced this to a single shape functional. Grad's number of $\Omega$ is

$G(\Omega)\ =\ \frac{1}{2|\Omega|}\ \inf_{\Sigma\in UA_N}\ \inf_{v\in V_\Sigma}\ \|\nabla^{\mathrm{sym}}v\|_{L^2(\Omega)}^2,$

where $UA_N$ is the set of antisymmetric $N\times N$ matrices of unit Frobenius (Hilbert–Schmidt) norm, $|\Omega|$ is the Lebesgue measure of $\Omega$, and $V_\Sigma$ is the set of divergence-free $H^1$ vector fields on $\Omega$ with $\nabla^a v\equiv\Sigma$ in $\Omega$ and $v\cdot n=0$ on $\partial\Omega$. (The name goes back to a formula in Grad's 1965 analysis of Boltzmann's H-theorem.) They proved: (i) the inverse tangency-Korn constant is bounded in terms of $G(\Omega)^{-1}$, the standard Korn constant, and a homology/Hodge constant $C_H(\Omega)$, a constant coming from the Hodge decomposition of vector fields into gradient, curl, and harmonic parts (equal to $1$ for convex $\Omega$), so the geometric content is concentrated in $G$; (ii) $G(\Omega)>0$ if and only if $\Omega$ is non-axisymmetric, in every dimension $N\ge 2$; (iii) quantitative lower bounds on $G(\Omega)$: general bounds in terms of Wasserstein distances (the optimal-transport distances $W_2$) between the uniform probability measure on $\Omega$ and its images under one-parameter families of rigid motions, valid in all dimensions, but more explicit "symmetrized" bounds only for $N=2$ and $N=3$.

Figalli sharpened the quantitative picture in dimensions two and three. Introducing the trace constant $T(\Omega)$ — an isoperimetric-type constant controlling $\int_{\partial\Omega}|f-c|\,d\mathcal{H}^{N-1}$ by $\int_{\Omega}|\nabla f|$ for the best constant $c$, where $\mathcal{H}^{N-1}$ is the $(N-1)$-dimensional Hausdorff measure — and the quadratic oscillation $\mathrm{osc}_2(\Gamma)=\mathrm{osc}_{x\in\Gamma}|x|^2$ of a boundary curve $\Gamma$ (with $\mathrm{osc}_\Gamma f=\sup_\Gamma f-\inf_\Gamma f$; for a whole boundary, $\mathrm{osc}_2(\partial\Omega)$ is the sum of the oscillations over the connected components of $\partial\Omega$), he proved for $N=2$

$G(\Omega)\ \ge\ \frac{1}{32\,|\Omega|^2\,T(\Omega)^2}\ \Big(\inf_{x_0\in\mathbb{R}^2}\ \mathrm{osc}_2(x_0+\partial\Omega)\Big)^{2},$

with a stronger version, under connected boundary, in terms of the Hausdorff distance (the distance between two sets, $d_H(A,B)=\max\{\sup_{a\in A}\mathrm{dist}(a,B),\ \sup_{b\in B}\mathrm{dist}(b,A)\}$) between $\partial\Omega$ and a circle; and for $N=3$ an analogous bound in terms of $\inf_{x_0,\omega}\int_{\mathbb{R}}\mathrm{osc}_2(x_0+\partial\Omega_{\omega,t})\,dt$, where $\partial\Omega_{\omega,t}=\partial\Omega\cap\{\omega\cdot x=t\}$ are the plane slices of the boundary orthogonal to the rotation axis $\omega$, with a stronger form for convex $\Omega$ involving Hausdorff distances of the slices to circles. These bounds are elementary, computable, and strictly stronger than the earlier ones on explicit examples (for a smoothed disc with a thin $\varepsilon$-strip removed they give $G(\Omega)\ge c$ where the Desvillettes–Villani bounds give $c\,\varepsilon^2$).

Figalli's proof reduces, via the trace inequality, to lower-bounding the boundary integral $\int_{\partial\Omega}|(\Sigma x)\cdot n|\,d\mathcal{H}^{N-1}$ by oscillation quantities. The two- and three-dimensional arguments lean on the special structure of antisymmetric matrices in low dimension: for $N=2$ the matrix $\Sigma$ is essentially unique (a quarter-turn), and for $N=3$ every $\Sigma\in UA_3$ is $x\mapsto\omega\wedge x/\sqrt{2}$ for a single unit axis $\omega$, so the coarea formula (the standard layer-cake decomposition of a surface integral by the level sets of a function) splits the boundary integral into two-dimensional computations on the slices $\partial\Omega_{\omega,t}$. In $N\ge 4$ this structure disappears: the normal form of an antisymmetric matrix is a simultaneous rotation with generically distinct angular weights in $\lfloor N/2\rfloor$ mutually orthogonal 2-planes, so there is no single axis and no canonical slicing. Figalli's footnote states that only $N=2,3$ are treated — "which are the most interesting from a physical point of view" — and that "it could be possible that our techniques can be employed also in the case $N\ge 4$"; the extension has never been carried out.

Problem Statement

Establish explicit geometric lower bounds on Grad's number in dimensions four and higher. Concretely: for bounded domains $\Omega\subset\mathbb{R}^N$ of at least $C^1$ regularity, for arbitrary $N\ge 4$, that are non-axisymmetric in the sense of Desvillettes–Villani (not invariant under any one-parameter group $(e^{tR})_{t\in\mathbb{R}}$ of isometries with nonzero antisymmetric linear part), define explicitly computable geometric functionals of $\Omega$ that quantify its deviation from axisymmetry — analogous, in dimensions two and three, to Figalli's trace constant $T(\Omega)$ combined with the quadratic oscillation $\mathrm{osc}_2$ of boundary components or of boundary slices, or to Hausdorff distances from circles or surfaces of revolution — and prove an inequality of the form

$G(\Omega)\ \ge\ \Phi\big(F_1(\Omega),\dots,F_k(\Omega);N\big)$

with $\Phi$ explicit and its constants tracked, valid for all such $\Omega$. The bound should meet the standard of the known low-dimensional results: the functionals should be elementary geometric quantities of $\Omega$ of the Figalli type (trace constants, quadratic oscillations of boundary components or of plane slices, Hausdorff distances to circles or surfaces of revolution, or comparably elementary boundary or slice quantities) rather than a restatement of $G(\Omega)$ itself, of the all-dimension Wasserstein bounds of Desvillettes–Villani, or of the boundary integral that the two- and three-dimensional proofs reduce to bounding, and the estimate should be demonstrably non-vacuous and of the correct order on near-axisymmetric model domains — for example, elongated ellipsoids in $\mathbb{R}^4$ or a ball with a thin slab removed, the higher-dimensional analogues of the examples on which the two- and three-dimensional bounds of Figalli and Desvillettes–Villani are compared. The techniques are unconstrained: the source remark concerns whether Figalli's approach extends, but any method yielding an explicit geometric lower bound that meets this standard in dimensions $N\ge 4$ answers the question. The first open case is $N=4$, and the source question concerns the extension to all $N\ge 4$.

References

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