[Paper Review] The sharp affine $L^2$ Sobolev trace inequality and variants

Pablo L. De Nápoli, Julián Haddad+2 more|arXiv (Cornell University)|Dec 15, 2015

Nonlinear Partial Differential Equations18 references3 citations

TL;DR

This paper establishes a sharp affine $L^2$ Sobolev trace inequality on the half-space $\mathbb{R}^n_+$ using the $L_p$ Busemann-Petty centroid inequality, which is stronger than and implies the classical sharp $L^2$ Sobolev trace inequality of Escobar and Beckner. The new inequality is invariant under affine transformations and characterizes equality cases, revealing a deeper geometric structure independent of Euclidean norms.

ABSTRACT

We establish a sharp affine $L^p$ Sobolev trace inequality by using the $L_p$ Busemann-Petty centroid inequality. For $p = 2$, our affine version is stronger than the famous sharp $L^2$ Sobolev trace inequality proved independently by Escobar and Beckner. Our approach allows also to characterize all cases of equality in this case. For this new inequality, no Euclidean geometric structure is needed.

Motivation & Objective

• To establish a sharp affine $L^2$ Sobolev trace inequality that is stronger than the classical sharp $L^2$ Sobolev trace inequality.
• To demonstrate that the new inequality is invariant under all affine transformations of $\mathbb{R}^n_+$, unlike the classical version which depends on Euclidean geometry.
• To characterize all cases of equality in the new inequality, extending the known extremal functions of Escobar and Beckner.
• To show that the new inequality does not rely on the choice of norm in $\mathbb{R}^n$, depending only on vector space structure and Lebesgue measure.
• To bridge affine isoperimetric inequalities with trace inequalities by using the $L_p$ Busemann-Petty centroid inequality as a key tool.

Proposed method

• The authors use the $L_p$ Busemann-Petty centroid inequality as the central analytical tool to derive the new affine trace inequality.
• They apply a duality argument and Fubini's theorem to relate the $L^p$-norm of the gradient to the trace on the boundary $\partial\mathbb{R}^n_+$.
• The proof involves constructing a functional $\mathcal{E}_p(f)$ that controls the trace norm via the $L_p$ centroid body of the function's level sets.
• The inequality is derived by combining Hölder’s inequality, Young’s inequality, and the $L_p$ Busemann-Petty centroid inequality to bound the trace in terms of the total variation of the gradient.
• The authors define a convex function $C_f^*$ on $\mathbb{R}^n$ that generalizes the Euclidean norm, enabling affine invariance.
• A detailed asymptotic analysis and beta function evaluation are used to compute the sharp constant in the inequality, relying on gamma function identities.

Experimental results

Research questions

• RQ1 Can a sharp affine version of the $L^2$ Sobolev trace inequality be derived that is stronger than the classical sharp $L^2$ Sobolev trace inequality of Escobar and Beckner?
• RQ2 What is the role of the $L_p$ Busemann-Petty centroid inequality in constructing affine-invariant trace inequalities?
• RQ3 How do the extremal functions of the new affine inequality relate to those of the classical inequality, and what characterizes equality in the new setting?
• RQ4 Can the new inequality be formulated independently of the Euclidean norm, relying only on vector space and measure-theoretic structure?
• RQ5 Is the new inequality strictly stronger than the classical one, and if so, under what conditions do they differ in extremal functions?

Key findings

• The sharp affine $L^2$ Sobolev trace inequality is strictly stronger than the classical sharp $L^2$ Sobolev trace inequality of Escobar and Beckner, and implies it as a special case.
• The new inequality is invariant under all affine transformations of $\mathbb{R}^n_+$, making it independent of the choice of norm in $\mathbb{R}^n$, a property absent in the classical version.
• All cases of equality in the new inequality are characterized: they occur precisely when the level set $K_{f,0}$ of the function is an ellipsoid, and after an appropriate $\mathrm{GL}_{n,+}$ transformation, the function is radial in $x$.
• The extremal functions of the new inequality are exactly those of the classical inequality, i.e., functions of the form $f(t,x) = \gamma \left( (t+\delta)^2 + |x - x_0|^2 \right)^{-(n-2)/2}$, confirming consistency with known results.
• The sharp constant in the inequality is computed explicitly using gamma functions and beta integrals, and depends only on $n$ and $p=2$, with the expression matching known values when restricted to the classical case.
• The method provides a framework to extend the result to general $1 < p < n$, though full characterization of extremal functions for $p \neq 2$ remains open.