2023/12/04 by Yingfang Zhang, Yuxuan Zhou, Zhang, Yingfang +3 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Partial Differential Equations #Numerical methods in engineering
paper · pdf · doi:10.48550/arxiv.2312.01766
openalex publication_date 2023/12/04 · openalex created_date 2023/12/06 · openalex updated_date 2026/07/28
In this paper, we study the stability of fractional Sobolev trace inequality within both the functional and critical point settings. In the functional setting, we establish the following sharp estimate: CBE(n,m,α)inf_v\inMn,m,α\Vert f-v\VertDα(ℝn)2 ≤ \Vert f\VertDα(ℝn)2 - S(n,m,α) \Vertτmf\VertLq(ℝn-m)2, where 0≤ m< n, (m)/(2)<α<(n)/(2), q=(2(n-m))/(n-2α) and Mn,m,α denotes the manifold of extremal functions. Additionally, We find an explicit bound for the stability constant CBE and establish a compactness result ensuring the existence of minimizers. In the critical point setting, we investigate the validity of a sharp quantitative profile decomposition related to the Escobar trace inequality and establish a qualitative profile decomposition for the critical elliptic equation Δu= 0 \quadin ℝ+n, (∂ u)/(∂ t)=-|u|(2)/(n-2)u \quadon ∂ℝ+n. We then derive the sharp stability estimate: CCP(n,ν)d(u,MEν)≤ \Vert Δu +|u|(2)/(n-2)u\VertH-1(ℝ+n), where ν=1,n≥ 3 or ν≥2,n=3 and MEν represents the manifold consisting of ν weak-interacting Escobar bubbles. Through some refined estimates, we also give a strict upper bound for CCP(n,1), which is (2)/(n+2).