Calculus of Variations and Geometric Measure Theory

A. Ponce - D. Spector

A Boxing Inequality for the Fractional Perimeter

created by spector on 18 Mar 2017



Inserted: 18 mar 2017
Last Updated: 18 mar 2017

Pages: 32
Year: 2017


We prove the Boxing inequality: \[ \mathcal{H}^{d-\alpha}_\infty(U) \leq C\alpha(1-\alpha)\int_U \int_{\mathbb{R}^{d}\setminus U} \frac{\mathrm{d}y \, \mathrm{d}z}{\mid y-z\mid^{\alpha+d}}, \] for every $\alpha \in (0,1)$ and every bounded open subset $U \subset \mathbb{R}^d$, where $\mathcal{H}^{d-\alpha}_\infty(U)$ is the Hausdorff content of $U$ of dimension \(d -\alpha\) and the constant $C > 0$ depends only on $d$. We then show how this estimate implies a trace inequality in the fractional Sobolev space $W^{\alpha, 1}(\mathbb{R}^d)$ that includes Sobolev's $L^{\frac{d}{d - \alpha}}$ embedding, its Lorentz-space improvement, and Hardy's inequality. All these estimates are thus obtained with the appropriate asymptotics as $\alpha$ tends to \(0\) and \(1\), recovering in particular the classical inequalities of first order. Their counterparts in the full range \(\alpha \in (0, d)\) are also investigated.