Suppose \(\Omega\subseteq \mathbb{R}^d\) is bounded with positive inradius \(r>0\). Let \( f\) be measurable with \((\supp f)_\varepsilon\subseteq \Omega\), where \((\supp f)_\varepsilon\) denotes the \(\varepsilon\)-neighbourhood of \(\supp f\).
There exists positive constants \(c\), \(C\) depending only on \(d\), such that
\begin{equation}\label{eq:1} [f]_{H^{1/2}(\Omega)}^2 + \Bigl(1+\frac{\diam \Omega}{\varepsilon}\Bigr)^{-(d+1)}\frac{c}{\diam \Omega} \int_{\Omega} |f|^2 \leq [f]_{H^{1/2}(\mathbb{R}^d)}^2 \leq [f]_{H^{1/2}(\Omega)}^2 + \Bigl(1+\frac{\diam \Omega}{\varepsilon}\Bigr)^{d+1}\frac{C}{r} \int_{\Omega} |f|^2. \end{equation}- For the case that \(\Omega\) is a cube, this statement is proven in [1, Lemma 5.1]. There, \(\varepsilon\) is a multiplicative of the side length of the cube. This simplifies the expressions involving \(\frac{\diam \Omega}{\varepsilon}\). Furthermore, the authors assumed \(f\in C^\infty (\mathbb{R}^d)\), but this assumption can be relaxed.
- This result implies, if \(f\) is compactly supported, then, for suitable \(\Omega\subseteq \mathbb{R}^d\) with \(\supp f\subseteq \Omega\), \(f\in H^{1/2}(\mathbb{R}^d)\) if and only if \(f\in H^{1/2}(\Omega)\).
By partitioning the domain, we find
\begin{equation}\label{eq:2} [f]^2_{H^{1/2}(\mathbb{R}^d)}=[f]^2_{H^{1/2}(\Omega)}+I, \end{equation}where
\[ I=2\int_{R^d\setminus \Omega} \int_{\Omega} \frac{\lvert f(x)-f(y)\rvert^2}{\lvert x-y\rvert^{d+1}} \,dy \,dx =2\int_{R^d\setminus \Omega} \int_{\supp f} \frac{\lvert f(y)\rvert^2}{\lvert x-y\rvert^{d+1}} \,dy \,dx. \]Let \(p\in \Omega\) be the center of an inscribed ball with radius \(r\). Using (0x68d03834) , we deduce for \(x\in \mathbb{R}^d\setminus \Omega\) and \(y\in \supp f\)
\[ C^{-1} \lvert x-p\rvert \le \lvert x-y\rvert \le C \lvert x-p\rvert, \]where \(C=(1+\tfrac{\diam \Omega}{\varepsilon})\). By applying Fubini’s theorem and radial integration formula , we then obtain
\[ \Bigl(1+\frac{\diam \Omega}{\varepsilon}\Bigr)^{-(d+1)}\frac{c'}{\diam \Omega }\int_{\Omega} \lvert f(x)\rvert^2 \,dx \le I\le \Bigl(1+\frac{\diam \Omega}{\varepsilon}\Bigr)^{d+1}\frac{C'}{r} \int_{\Omega} \lvert f(x)\rvert^2 \,dx , \]where \(c'\) and \(C'\) are constants depending only on \(d\).
Together, this inequality and \eqref{eq:2} yield \eqref{eq:1}.
See also Link to heading
References Link to heading
- M. Di Cristo, E. Francini, C. Lin, S. Vessella, and J. Wang,
Carleman estimate for second order elliptic equations with Lipschitz leading coefficients and jumps at an interface,
Journal de Mathématiques Pures et Appliquées, vol. 108, no. 2, pp. 163–206, 2017. doi:10.1016/j.matpur.2016.10.015