The isoperimetric problem in the Grushin space \(\mathbb{R}^{h+1}\) with density \(|x|^p\)
The authors consider the Grushin space \(\mathbb{R}^{h+1} = \{(x, y) \mid x \in \mathbb{R}^h, y \in \mathbb{R}\}\) with density \(f = |x|^p\), \(p > -h + 1\). The weighted \(\alpha\) perimeter of a Lebesgue measurable set \(E \subset \mathbb{R}^{h + 1}\) is defined as follows: \[ P_{\alpha, f}(E) = \sup \left\{ \int_E \operatorname{div}_\alpha(|x|^p \varphi(x, y)) dx dy : \varphi \in C^1_c(\mathbb{R}^{h+1}; \mathbb{R}^{h+1}), \max | \varphi(x, y) | \leq 1 \right\}, \] where \(\operatorname{div}_\alpha \varphi = X_1 \varphi_1 + \cdots + X_h \varphi_h + Y \varphi_{h + 1}\), \(X_i = \frac{\partial }{\partial x_i}\), \(i=\overline{1, h}\), \(Y = \frac{\partial }{\partial y}\). The weighted volume of the set \(E\) is defined as \(V_f(E) = \int_E |x|^p dx dy\). The goal of the paper is to study the existence and characterization of minimizers of the problem \[ \inf \left\{ P_{\alpha, f} (E) \mid V_f(E)=v_0 \right\} \] for \(x\)-spherically symmetric sets \(E\). From the abstract: ``First we prove the existence of weighted isoperimetric sets. Then we deduce that, up to a vertical translation, a dilation and a negligible set, the weighted isoperimetric set is only of the form: \(\left\{ (x, y) \in \mathbb{R}^{h + 1} : |y| < \int_{\arcsin |x|}^{\frac{\pi}{2}} \sin^{\alpha + 1}(t) \, dt, |x| < 1 \right\}\).'' The paper contains an extensive bibliography on the topic, including [\textit{N. Garofalo} and \textit{D.-M. Nhieu}, Commun. Pure Appl. Math. 49, No. 10, 1081--1144 (1996; Zbl 0880.35032)] (the general definition and study of the perimeter of a Lebesgue measurable set \(E \subset \mathbb{R}^n\) defined via a system of self-adjoint vector fields) and [\textit{R. Monti} and \textit{D. Morbidelli}, J. Geom. Anal. 14, No. 2, 355--368 (2004; Zbl 1076.53035)] (the isoperimetric problem for the Grushin plane).
scientific article; zbMATH DE number 6898048
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