Isoperimetric inequality in the Grushin plane
The so-called Grushin plane is a plane \(\mathbb R^2\) with ``Riemannian'' metric \(ds^2= dx^2+ |x|^{-2\alpha}dy^2\) for a constant \(\alpha\geq 0\), which is one of the simplest examples of sub-Riemannian geometry. In the more general case of Carnot-Carathéodory spaces, isoperimetric inequalities are discussed by Gromov and the others. But in all previous results there are no sharp constants obtained. In this paper the authors prove a sharp isometric inequality in the Grushin plane, precisely they give the following Theorem 1: Let \(\alpha\geq 0\). For any measurable set \(E\subset \mathbb R^2\) with finite measure \(|E|\), define its \(\alpha\)-perimeter as \[ P\alpha(E)= \sup\biggl\{\int_E(\partial_x\varphi_1+ |x|^{\alpha}\partial_y\varphi_2)\,dx\,dy: \varphi_1,\varphi_2\in C^1_0 (\mathbb R^2),\;\sup_{\mathbb R^2}(\varphi_1^2+ \varphi_2^2)^{1/2}\leq 1\biggr\}. \] Then we have \[ |E|\leq\frac{\alpha+ 1} {\alpha+2}\Biggl(2\int_0^{\pi}\sin^\alpha t\,dt\Biggr)^{-\frac{1} {\alpha+1}}P_{\alpha}(E)^{\frac{\alpha+2}{\alpha+1}}. \] Equality holds for the isoperimetric set \[ E_{\alpha}= \Biggr\{(x,y)\in \mathbb R^2: |y| < \int_{\arcsin|x|}^{\pi/2}\sin^{\alpha+ 1}t\,dt,|x|< 1\Biggr\}. \] If \(\alpha> 0\), isoperimetric sets are unique up to vertical translations and dilations of the form \((x,y)\to \delta_{\lambda}(x, y)= (\lambda x, \lambda^{\alpha+ 1}y)\), \(\lambda> 0\). From Theorem 1 we can get a Gagliardo-Nirenberg inequality for the Grushin sub-elliptic gradient, i.e. \[ \biggl(\int_{\mathbb R^2}|f|^{\frac{d}{d- 1}}dx\,dy\biggr)^{\frac{d- 1}{d}} \leq c(\alpha)^{\frac{d-1}{d}} \int_{\mathbb R^2}(|\partial_x f|^2+ |x|^{2\alpha} |\partial_yf|^2)^{1/2}\,dx\,dy \] for all functions \(f\in C^\infty_0(\mathbb R^2)\), where \(d= \alpha+ 2\) and \(c(\alpha)= \frac{\alpha+1} {\alpha+ 2}(2\int_ 0^{\pi}\sin^{\alpha}t\,dt)^{-\frac{1} {\alpha+ 1}}\).
scientific article; zbMATH DE number 2147106
scientific article