Stability of the Sobolev--Escobar bridge inequality

We study the local stability of the bridge family $$\mathrm{\Phi }(T):=\underset{u\in {\mathcal{A}}_T}{\inf }\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathbb{R}}_{+}^n)},T>0,n\ge 3,$$ where $${\mathcal{A}}_T:=\{u\in {\dot{H}}^1({\mathbb{R}}_{+}^n):\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\frac{2n}{n-2}}({\mathbb{R}}_{+}^n)}=1,\mathrm{\parallel }u{\mathrm{\parallel }}_{L^{\frac{2(n-1)}{n-2}}(\mathrm{\partial }{\mathbb{R}}_{+}^n)}=T\},$$ and ${\dot{H}}^1({\mathbb{R}}_{+}^n)$ is the completion of $C_c^{\mathrm{\infty }}(\overline{{\mathbb{R}}_{+}^n})$ in the norm $\mathrm{\parallel }\mathrm{\nabla }\phi {\mathrm{\parallel }}_{L^2({\mathbb{R}}_{+}^n)}$. Let ${\mathcal{M}}_T$ denote the set of minimizers of $\mathrm{\Phi }(T)$. We prove that, for every $T\mathrm{\ne }{T}_E$, there exists ${\alpha }_T>0$ such that $$\mathrm{\parallel }\mathrm{\nabla }u{\mathrm{\parallel }}_{L^2({\mathbb{R}}_{+}^n)}^2-\mathrm{\Phi }(T{)}^2\ge {\alpha }_T{d}_T(u,{\mathcal{M}}_T{)}^2+o\text{\hspace{0.17em}⁣}(d_T(u,{\mathcal{M}}_T{)}^2)\text{for all }u\in {\mathcal{A}}_T,$$ where $T_E$ is the Escobar threshold and $d_T$ is the distance in ${\dot{H}}^1({\mathbb{R}}_{+}^n)$.{for all }u\in\mathcal A_T, ] where $T_E$ is the Escobar threshold and $d_T$ is the distance in ${\dot{H}}^1({\mathbb{R}}_{+}^n)$.