A note on the Sobolev trace inequality

Abstract:

Consider the classical Sobolev trace inequality \begin{equation*} \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)} \geq K\|\varphi \|_{L^{\frac {2(n-1)}{n-2}}(\partial \mathbb {R}^n_+)} \end{equation*} for all $\varphi \in W^{1,2}_0(\mathbb {R}^n_+)$, where $K$ is the best constant. Here, $W^{1,2}_0(\mathbb {R}^n_+)$ is the space obtained by taking the completion in the norm $\|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)}$ of the set of all smooth functions with support contained in the closure of $\mathbb {R}^n_+$, and $n\geq 3$. Let $\mathcal {M}$ be the set of functions for which we have equality in the Sobolev trace inequality above. In this note, we show that there is a positive constant $\alpha$ such that \begin{equation*} \|\nabla \varphi \|_{L^2(\mathbb {R}^n_+)}^2-K^2 \|\varphi \|_{L^{\frac {2(n-1)}{n-2}}(\partial \mathbb {R}^n_+)}^2\geq \alpha d(\varphi ,\mathcal {M})^2 \end{equation*} for all $\varphi \in W^{1,2}_0(\mathbb {R}^n_+)$, where $d$ is the distance in the Sobolev space $W^{1,2}_0(\mathbb {R}^n_+)$.