Existence of optimizers for the sharp stability constant in the logarithmic Sobolev inequality
We study the sharp stability constant in the logarithmic Sobolev inequality, defined as the infimum of the logarithmic Sobolev deficit divided by the squared $L^2$-distance from the manifold of Gaussian optimizers. For every $N\geq 1$, first we prove that this infimum is attained in the Euclidean formulation of the inequality. Then, we show that every such optimizer decays exponentially fast at infinity, and therefore gives rise to an optimizer in the Gaussian formulation, where the same sharp constant appears. The proof is based on an extension of the general strategy introduced by Bianchi and Egnell for the Sobolev inequality and recently developed by König to prove the existence of optimizers for the corresponding sharp stability constant. The main difference in the logarithmic Sobolev setting is the analysis of minimizing sequences approaching the manifold of optimizers, since the entropy term $u^2\log u^2$ does not allow for a direct global second-order expansion. This is overcome by a perturbative estimate near the Gaussian manifold, which provides the compactness threshold needed to rule out the loss of compactness.