2022/09/09 by Wei, Juncheng, Wu, Yuanze · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2209.04118
In this paper, we consider the Euclidean logarithmic Sobolev inequality ∫ℝd|u|2log|u|dx≤(d)/(4)log((2)/(πd e)‖∇ u‖L2(ℝd)2), where u∈ W1,2(ℝd) with d≥2 and ‖u‖L2(ℝd)=1. It is well known that extremal functions of this inequality are precisely the Gaussians \mathfrakgσ,z(x)=(πσ)-(d)/(2)\mathfrakg*(√\fracσ2(x-z))\quadwith \mathfrakg*(x)=e-(|x|2)/(2). We prove that if u≥0 satisfying (ν-\frac12)c0<‖u‖H1(ℝd)2<(ν+\frac12)c0 and ‖-Δu+u-2ulog |u|‖H-1≤δ, where c0=‖\mathfrakg1,0‖H1(ℝd)2, ν∈ ℕ and δ>0 sufficiently small, then distH1(u, Mν)\lesssim‖-Δu+u-2ulog |u|‖H-1 which is optimal in the sense that the order of the right hand side is sharp, where Mν=\(\mathfrakg1,0(⋅-z1), \mathfrakg1,0(⋅-z2), ⋯, \mathfrakg1,0(⋅-zν))| zi∈\bbrd\. Our result provides an optimal stability of the Euclidean logarithmic Sobolev inequality in the critical point setting.