Qualitative stability for a family of trace Sobolev inequalities

Technical Summary: Qualitative Stability for a Family of Trace Sobolev Inequalities

Problem Statement
This paper addresses the qualitative stability of a one-parameter family of trace Sobolev inequalities on the half-space $H = \{x \in \mathbb{R}^n : x \cdot e_1 > 0\}$, where $n \ge 2$ and $p \in (1, n)$. The inequalities, established by Maggi and Neumayer [MN17], relate the $L^p$ norm of the gradient of a function $u$ to its $L^{p^*}$ norm in the interior and its $L^{p^\sharp}$ norm on the boundary $\partial H$. Specifically, for a fixed trace parameter $T \ge 0$, the sharp constant $\Phi_H(T)$ is defined via the variational problem:
$$\Phi_H(T) = \inf \{ \|\nabla u\|_{L^p(H)} : u \in \mathcal{A}_T \},$$
where $\mathcal{A}_T = \{ u \in \dot{W}^{1,p}(H) : \|u\|_{L^{p^*}(H)} = 1, \|u\|_{L^{p^\sharp}(\partial H)} = T \}$.
The extremals (minimizers) for this problem are fully characterized as specific dilations and translations of an explicit profile $U_T$. The central question is stability: if a sequence of functions $\{u_k\} \subset \mathcal{A}_T$ has a vanishing energy deficit $\delta_T(u_k) = \|\nabla u_k\|_{L^p}^p - \Phi_H(T)^p \to 0$, does the sequence converge (in the gradient norm) to the set of extremals $\mathcal{M}_T$?

While local quantitative stability was recently established for the case $p=2$ by Fan, Li, and Zhang [FLZ26], a global qualitative stability result (i.e., proving that vanishing deficit implies vanishing distance to extremals) was missing. This gap prevented the extension of the local quantitative result to a global one.

Methodology
The proof of qualitative stability relies on the concentration-compactness principle combined with a novel strict binding inequality for the energy functional.

1. Concentration-Compactness Analysis: The author adapts the standard concentration-compactness lemmas (Lions [Lio85]) to the trace setting. The proof proceeds by contradiction: assuming a sequence with vanishing deficit does not converge to an extremal, the sequence must either vanish (ruled out by normalization) or split (dichotomy) into two asymptotically non-interacting profiles.
2. The Strict Binding Inequality (Theorem 1.3): The core of the argument is ruling out the "splitting" scenario. The author proves that for any decomposition of the mass and trace norms into two parts $(m_1, t_1)$ and $(m_2, t_2)$, the sum of the energies of the separate profiles is strictly greater than the energy of the combined profile:
   $$\Phi_H(T)^p < m_1^p \Phi_H\left(\frac{t_1}{m_1}\right)^p + m_2^p \Phi_H\left(\frac{t_2}{m_2}\right)^p.$$
   This inequality is non-trivial because, unlike the classical Sobolev inequality on $\mathbb{R}^n$, the constant $\Phi_H(T)$ depends non-linearly on $T$, preventing simple scaling arguments from ruling out splitting.
3. Mass Transportation Argument: To prove the strict binding inequality, the paper utilizes the mass transportation proof of the trace inequality from [MN17]. The author constructs a specific test function $w$ consisting of two widely separated copies of extremal profiles. By analyzing the optimal transport map (Brenier map) associated with $w$, the author exploits the cyclical monotonicity of the transport map. The geometric separation of the profiles forces the transport map to behave in a way that is incompatible with the cyclical monotonicity condition unless the energy deficit is bounded away from zero. This yields a definite lower bound on the deficit, establishing the strict inequality.

Key Results

• Theorem 1.1 (Qualitative Stability): For any $p \in (1, n)$ and $T > 0$, if a sequence $\{u_k\} \subset \mathcal{A}_T$ satisfies $\delta_T(u_k) \to 0$, then the distance to the set of extremals $d_T(u_k) \to 0$. This confirms that the only way to approach the sharp constant is by approaching the family of extremals.
• Theorem 1.3 (Strict Binding Inequality): The strict inequality relating the energy of a split configuration to the combined energy is proven for all $T > 0$.
• Corollary 1.2 (Global Quantitative Stability for $p=2$): By combining the new qualitative result (Theorem 1.1) with the existing local quantitative analysis from [FLZ26], the paper establishes global quantitative stability for the case $p=2$. Specifically, there exists a constant $\alpha'_T > 0$ such that $\delta_T(u) \ge \alpha'_T d_T(u)^2$ for all $u \in \mathcal{A}_T$.

Significance and Claims
The paper claims to resolve an open problem raised in [FLZ26] regarding the global quantitative stability of trace Sobolev inequalities for $p=2$. The author emphasizes that the primary obstruction to this result was the lack of a qualitative stability theorem, which this note provides.

The work is presented as a "short note" filling a specific theoretical gap. The author modestly notes that while the mass transportation approach is powerful, it faces difficulties in providing direct quantitative estimates for the distance to extremals (as done for the isoperimetric inequality in other works) due to the degeneracy of control in regions where the function is small. Consequently, the paper relies on the qualitative strict binding inequality derived via mass transportation to enable the global quantitative result.

The results extend the understanding of stability for functional inequalities where the optimal constant depends non-trivially on a parameter, a setting where standard homogeneity arguments fail. The paper does not propose new applications or experimental proposals but strictly focuses on the rigorous analysis of the stability properties of these specific inequalities.