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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.12677 |
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| _version_ | 1866908962912206848 |
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| author | Song, Fan Gui-Dong, Li Jianjun, Zhang |
| author_facet | Song, Fan Gui-Dong, Li Jianjun, Zhang |
| contents | In this note, we study the local stability of the bridge family \[ Φ(T):=\inf_{u\in\mathcal A_T}\|\nabla u\|_{L^2(\mathbb R^n_+)}, \qquad T>0,\quad n\ge3, \] 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 \(\|\nabla φ\|_{L^2(\mathbb R^n_+)}\). Let \(\mathcal M_T\) denote the set of minimizers of \(Φ(T)\). We prove that, for every \(T\neq T_E\), there exists \(α_T>0\) such that \[ \|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-Φ(T)^2 \ge α_T\,d_T(u,\mathcal M_T)^2 +o\!\bigl(d_T(u,\mathcal M_T)^2\bigr) \qquad\text{for all }u\in\mathcal A_T, \] where \(T_E\) is the Escobar threshold and \(d_T\) is the distance in \(\dot H^1(\mathbb R^n_+)\). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_12677 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on the Sobolev--Escobar bridge inequality Song, Fan Gui-Dong, Li Jianjun, Zhang Analysis of PDEs In this note, we study the local stability of the bridge family \[ Φ(T):=\inf_{u\in\mathcal A_T}\|\nabla u\|_{L^2(\mathbb R^n_+)}, \qquad T>0,\quad n\ge3, \] 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 \(\|\nabla φ\|_{L^2(\mathbb R^n_+)}\). Let \(\mathcal M_T\) denote the set of minimizers of \(Φ(T)\). We prove that, for every \(T\neq T_E\), there exists \(α_T>0\) such that \[ \|\nabla u\|_{L^2(\mathbb{R}_{+}^n)}^2-Φ(T)^2 \ge α_T\,d_T(u,\mathcal M_T)^2 +o\!\bigl(d_T(u,\mathcal M_T)^2\bigr) \qquad\text{for all }u\in\mathcal A_T, \] where \(T_E\) is the Escobar threshold and \(d_T\) is the distance in \(\dot H^1(\mathbb R^n_+)\). |
| title | A note on the Sobolev--Escobar bridge inequality |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2604.12677 |