On the stability of fractional Sobolev trace inequality and corresponding profile decomposition (2312.01766v2)

Abstract: 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: $$C_{\mathrm{BE}}(n,m,\alpha)\inf_{v\in\mathcal{M}{n,m,\alpha}}\left\Vert f-v\right\Vert{D_\alpha(\mathbb{R}^n)}^2 \leq \left\Vert f\right\Vert_{D_\alpha(\mathbb{R}^n)}^2 - S(n,m,\alpha) \left\Vert\tau_mf\right\Vert_{L^{q}(\mathbb{R}^{n-m})}^2,$$ where $0\leq m< n$, $\frac{m}{2}<\alpha<\frac{n}{2}, q=\frac{2(n-m)}{n-2\alpha}$ and $\mathcal{M}{n,m,\alpha}$ denotes the manifold of extremal functions. Additionally, We find an explicit bound for the stability constant $C{\mathrm{BE}}$ 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 \begin{equation*} \Delta u= 0 \quad\text{in }\mathbb{R}+^n,\quad\frac{\partial u}{\partial t}=-|u|^{\frac{2}{n-2}}u \quad\text{on }\partial\mathbb{R}+^n. \end{equation*} We then derive the sharp stability estimate: $$ C_{\mathrm{CP}}(n,\nu)d(u,\mathcal{M}{\mathrm{E}}^{\nu})\leq \left\Vert \Delta u +|u|^{\frac{2}{n-2}}u\right\Vert{H^{-1}(\mathbb{R}_+^n)}, $$ where $\nu=1,n\geq 3$ or $\nu\geq2,n=3$ and $\mathcal{M}{\mathrm{E}}^\nu$ represents the manifold consisting of $\nu$ weak-interacting Escobar bubbles. Through some refined estimates, we also give a strict upper bound for $C{\mathrm{CP}}(n,1)$, which is $\frac{2}{n+2}$.