[Paper Review] Stability for the Sobolev inequality: existence of a minimizer

Tobias König|arXiv (Cornell University)|Nov 25, 2022

Nonlinear Partial Differential Equations4 citations

TL;DR

This paper establishes the existence of a minimizer for the Bianchi-Egnell stability inequality associated with the Sobolev inequality in dimensions $d \geq 3$, using a refined Brezis-Lieb-type compactness argument. It proves the strict inequality $c_{\text{BE}} < 2 - 2^{\frac{d-2}{d}}$, ruling out two-bubble configurations as minimizers, and extends the result to the fractional Sobolev case for $s \in (0, d/2)$, $d \geq 2$. The key contribution is the existence of an optimizer for the stability constant, resolving a long-standing open problem.

ABSTRACT

We prove that the stability inequality associated to Sobolev's inequality and its set of optimizers $\mathcal M$ and given by \[ \frac{\| abla f\|_{L^2(\mathbb R^d)}^2 - S_d \|f\|_{L^\frac{2d}{d-2}(\mathbb R^d)}^2}{ \inf_{h \in \mathcal M} \| abla (f - h)\|_{L^2(\mathbb R^d)}^2 } \geq c_{BE} &gt; 0 \qquad ext{ for every } f \in \dot{H}^1(\mathbb R^d),\] which is due to Bianchi and Egnell, admits a minimizer for every $d \geq 3$. Our proof consists in an appropriate refinement of a classical strategy going back to Brezis and Lieb. As a crucial ingredient, we establish the strict inequality $c_{BE} &lt; 2 - 2^\frac{d-2}{d}$, which means that a sequence of two asymptotically non-interacting bubbles cannot be minimizing. Our arguments cover in fact the analogous stability inequality for the fractional Sobolev inequality for arbitrary fractional exponent $s \in (0, d/2)$ and dimension $d \geq 2$.

Motivation & Objective

• To resolve the open problem of whether the best constant $c_{\text{BE}}$ in the Bianchi-Egnell stability inequality for the Sobolev inequality is achieved by a minimizer.
• To show that minimizing sequences for the Bianchi-Egnell quotient cannot concentrate into two asymptotically non-interacting Talenti bubbles, thereby excluding a non-compact alternative to minimizer existence.
• To extend the existence result to the fractional Sobolev setting for arbitrary $s \in (0, d/2)$ and $d \geq 2$, generalizing the classical case.
• To establish the strict inequality $c_{\text{BE}} < 2 - 2^{\frac{d-2}{d}}$, which rules out two-bubble configurations as potential minimizers.

Proposed method

• A refined version of the classical Brezis-Lieb concentration-compactness strategy is employed to analyze minimizing sequences for the Bianchi-Egnell quotient.
• The proof relies on a sharp asymptotic expansion of the functional involving two Talenti bubbles with diverging concentration rates and centers.
• The analysis shows that the quotient for two-bubble configurations converges to $2 - 2^{\frac{d-2}{d}}$, which is strictly greater than $c_{\text{BE}}$, thus excluding such configurations from being minimizing.
• The argument uses a spectral comparison to rule out one-bubble-type minimizing sequences, leveraging the previously known strict inequality $c_{\text{BE}} < \frac{4}{d+4}$.
• The method is generalized to fractional Sobolev spaces by adapting the functional framework and asymptotic expansions to the nonlocal Dirichlet energy $\|(-\Delta)^{s/2} f\|_{L^2}^2$.
• A key technical component is the computation of the first and second derivatives of a profile functional at $\mu = 1$, which quantifies the behavior of perturbations around the Talenti bubble.

Experimental results

Research questions

• RQ1 Does the best constant $c_{\text{BE}}$ in the Bianchi-Egnell stability inequality for the Sobolev inequality arise from a minimizer, or is it only attained in the limit?
• RQ2 Can two asymptotically non-interacting Talenti bubbles form a minimizing sequence for the Bianchi-Egnell quotient?
• RQ3 Is the strict inequality $c_{\text{BE}} < 2 - 2^{\frac{d-2}{d}}$ valid, which would rule out two-bubble configurations as minimizers?
• RQ4 Does the existence of a minimizer for the stability inequality extend to the fractional Sobolev case with $s \in (0, d/2)$?

Key findings

• The best constant $c_{\text{BE}}$ in the Bianchi-Egnell stability inequality is achieved by a minimizer for all dimensions $d \geq 3$, resolving a long-standing open problem.
• The strict inequality $c_{\text{BE}} < 2 - 2^{\frac{d-2}{d}}$ holds, which implies that two-bubble configurations cannot be minimizing sequences.
• The minimizer exists not only for the classical Sobolev case but also for the fractional Sobolev inequality with arbitrary $s \in (0, d/2)$ and $d \geq 2$, extending the result to a broader class of inequalities.
• The proof establishes that minimizing sequences cannot escape to infinity or split into two non-interacting bubbles, ensuring strong convergence to a minimizer.
• The analysis confirms that the spectral constant $c_{\text{BE}}^{\text{spec}} = \frac{4}{d+4}$ is strictly greater than $c_{\text{BE}}$, further ruling out one-bubble-type minimizers.
• The existence of a minimizer is established via a refined Brezis-Lieb argument combined with sharp asymptotic estimates on the functional behavior of two-bubble profiles.