Quantitative stability for the Trudinger-Moser inequality

Technical Summary: Quantitative Stability for the Trudinger–Moser Inequality

Problem Statement
The paper investigates the quantitative stability of the Trudinger–Moser inequality on smooth, bounded domains $\Omega \subset \mathbb{R}^2$. While the classical Sobolev embedding fails at the critical exponent $q=n$, the Trudinger–Moser inequality provides a substitute, asserting that functions in $W^{1,n}_0(\Omega)$ exhibit exponential integrability. Specifically, for $n=2$, the functional
$$E_{\alpha, \Omega}(u) = \int_\Omega e^{\alpha u^2} dx$$
is bounded for $\|\nabla u\|_{L^2(\Omega)} \leq 1$ if and only if $0 \leq \alpha \leq 4\pi$. The set of extremal functions (optimizers) $M_\alpha(\Omega)$ is known to be non-empty, but their precise analytical form remains unresolved even for the unit disk.

The central problem addressed is whether the "deficit" in the inequality—defined as the difference between the supremum $TM(\alpha, \Omega)$ and the functional value $E_{\alpha, \Omega}(u)$—controls the distance of a function $u$ to the set of optimizers $M_\alpha(\Omega)$. The authors seek to establish a quantitative stability estimate of the form:
$$\text{def}_{\alpha, \Omega}(u) \gtrsim \text{dist}_{\alpha, \Omega}(u)^2,$$
where the distance is measured in the $W^{1,2}_0$ norm (specifically the $L^2$ norm of the gradient).

Methodology
The proofs rely on spectral analysis of linearized operators around optimizers and a contradiction argument involving the asymptotic behavior of violating sequences.

1. Spectral Gap Analysis: The authors define two key eigenvalue problems associated with an optimizer $u^*$:

   • The linearized problem $(P_{\alpha, \Omega})$: $-\Delta \phi = \lambda (1 + 2\alpha (u^*)^2)e^{\alpha (u^*)^2} \phi$.
   • The nonlinear Trudinger–Moser problem $(TM_{\alpha, \Omega})$: $-\Delta v = \Lambda e^{\alpha (u^*)^2} v$.
     Let $\lambda_1(\alpha, \Omega)$ and $\Lambda_1(\alpha, \Omega)$ denote their respective first eigenvalues. A crucial step is establishing a spectral gap where $\Lambda_1(\alpha, \Omega)$ lies strictly between the first and second eigenvalues of the linearized problem, or strictly below the first non-radial eigenvalue in symmetric cases.

2. Contradiction Strategy: The proofs for the subcritical cases proceed by contradiction. Assuming the stability estimate fails, there exists a sequence of functions approaching the supremum but remaining far from the set of optimizers. By compactness (via Vitali's theorem and concentration-compactness), this sequence converges to an optimizer $u^*$. The normalized deviations satisfy an asymptotic Euler-Lagrange equation, converging to a limit function that must solve the linearized eigenvalue problem. The existence of a spectral gap renders this limit impossible (as it would imply a sign-definite eigenfunction orthogonal to the tangent space), thereby proving the stability estimate.

3. Extension from Sphere to Ball: Since the Trudinger–Moser functional is not homogeneous (unlike the Sobolev functional), the authors first establish stability on the unit sphere $S^1$ (where $\|\nabla u\|_{L^2}=1$) and then extend it to the closed unit ball $B_1$ using the monotonicity of the functional under radial rescalings.

4. Genericity via Transversality: To address the nondegenerate case, the authors employ a Sard–Smale transversality argument. By parameterizing the domain via diffeomorphisms and analyzing the augmented map of the Euler-Lagrange equation, they show that nondegeneracy (where the Hessian is coercive on the tangent space) holds for a generic set of parameters $(\alpha, \Omega)$.

Key Contributions and Results

The paper establishes three main theorems regarding the quadratic control of the distance to optimizers by the deficit:

• Theorem 1 (Small Subcritical Case): For any smooth bounded domain $\Omega$, there exists a threshold $\alpha_0 > 0$ such that for all $\alpha \in (0, \alpha_0)$, the stability estimate holds. This result relies on the fact that as $\alpha \to 0$, the eigenvalues of the linearized problems converge to the first Dirichlet eigenvalue of the Laplacian, creating a spectral gap.
• Theorem 2 (Nondegenerate Subcritical Case): For any $\alpha \in (0, 4\pi)$, if every optimizer is nondegenerate (i.e., the second variation is coercive on the tangent space), the stability estimate holds. The authors further prove that such nondegeneracy is a generic property, holding for an open and dense set of parameters $(\alpha, \Omega)$.
• Theorem 3 (Symmetric Critical Case): For the unit disk $D$ and the full critical range $\alpha \in (0, 4\pi]$, the stability estimate holds. This result is significant because it covers the critical exponent $\alpha = 4\pi$ without requiring smallness assumptions. The proof exploits the rotational symmetry of the disk, decomposing the linearized operator into angular Fourier modes. A strict spectral gap is shown between the radial eigenvalue and the first non-radial eigenvalue (introduced by a centrifugal term $r^{-2}$), ensuring coercivity in all directions.

Significance and Claims
The paper claims to provide the first quantitative stability estimates for the Trudinger–Moser inequality in the general setting where the precise form of extremals is unknown. Unlike previous stability results for inequalities like Sobolev or Yamabe, where the set of optimizers is explicitly known (often involving conformal transformations), the Trudinger–Moser extremals are not fully characterized. Consequently, the authors could not rely on explicit calculations of the distance to optimizers.

Instead, the authors introduce a new spectral gap inequality that serves as the core mechanism for their proofs. This gap allows them to bypass the need for explicit knowledge of the extremals, relying instead on the spectral properties of the linearized operator. The authors note that the spectral gap result itself may be of independent interest. The work extends the program of quantitative stability (initiated by Bianchi and Egnell for Sobolev inequalities) to a critical, non-homogeneous, and geometrically complex setting, demonstrating that quadratic stability holds even at the critical exponent in symmetric domains.