A note on the Sobolev--Escobar bridge inequality

In this note, we study the local stability of the bridge family $$\Phi(T):=\inf_{u\in\mathcal A_T}\|\nabla u\|_{L^2(\mathbb R^n_+)}, \qquad T>0,\quad n\ge3,$$ where $$\mathcal A_T := \Bigl\{ u\in \dot H^1(\mathbb R^n_+): \|u\|_{L^{\frac{2n}{n-2}}(\mathbb{R}_{+}^n)}=1,\ \|u\|_{L^{\frac{2(n-1)}{n-2}}(\partial\mathbb{R}_{+}^n)}=T \Bigr\},$$ and $\dot H^1(\mathbb R^n_+)$ is the completion of $C_c^\infty(\overline{\mathbb R^n_+})$ in the norm $\|\nabla \varphi\|_{L^2(\mathbb R^n_+)}$. Let $\mathcal M_T$ denote the set of minimizers of $\Phi(T)$. We prove that, for every $T\neq T_E$, there exists $\alpha_T>0$ such that $$\|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-\Phi(T)^2 \ge \alpha_T\,d_T(u,\mathcal M_T)^2 +o\!\bigl(d_T(u,\mathcal M_T)^2\bigr) \qquad\text{for all }u\in\mathcal A_T,$$ where $T_E$ is the Escobar threshold and $d_T$ is the distance in $\dot H^1(\mathbb R^n_+)$.