On the \(L_p\) torsional Minkowski problem for \(0 < p < 1\)
type: Item
instance of: scholarly article
title: On the \(L_p\) torsional Minkowski problem for \(0 < p < 1\)
zbMATH Open document ID: 1479.52008
author: Jinrong Hu
author: Jiaqian Liu
published in: Advances in Applied Mathematics
publication date: 2021-06-28T00:00:00Z
review text: Let \(\Omega\subseteq\mathbb{R}^n\) be a convex body (i.e., a convex, compact set with non-empty interior), and consider that \(u\) is the unique solution of the elliptic boundary-value problem \[ \begin{cases} \Delta u = -2 &\text{ in }\operatorname{int}(\Omega),\\ u = 0 &\text{ on }\partial\Omega. \end{cases} \] It is well-known that the torsional rigidity \(T(\Omega)\) of \(\Omega\) can be obtained from \(u\) as \[ T(\Omega) = \int_{\Omega}|\nabla u|^2\,dx. \] It is also known that \begin{align*} T(\Omega) &= \frac{1}{n+2}\int_{\partial\Omega}h(\Omega,g_{\Omega}(x))|\nabla u(x)|^2\,d\mathcal{H}^{n-1}(x) \\ &= \frac{1}{n+2}\int_{\mathbb{S}^{n-1}}h(\Omega,\theta)|\nabla u(g_{\Omega}^{-1}(\theta))|^2\,dS(\Omega,\theta), \end{align*} where \(h(\Omega,\cdot)\) is the usual support function of \(\Omega\), \(\mathcal{H}^{n-1}\) is the standard \((n-1)\)-dimensional Hausdorff measure, \(g_{\Omega}\) is the Gauss-map of \(\partial\Omega\) and \(S(\Omega,\cdot)\) is the surface area measure of \(\Omega\). This holds even if \(\Omega\) is non-smooth (in this case, it is known that the involved functions can be suitably regarded almost-everywhere). The torsion measure \(\mu^{\mathrm{tor}}(\Omega,\cdot)\) of \(\Omega\) is a Borel measure on \(\mathbb{S}^{n-1}\) defined as \begin{align*} \mu^{\mathrm{tor}}(\Omega,\eta) &= \int_{g_{\Omega}^{-1}(\eta)}|\nabla u(x)|^2\,d\mathcal{H}^{n-1}(x) \\ &= \int_{\eta}|\nabla u(g_{\Omega}^{-1}(\theta))|^2\,dS(\Omega,\theta) \end{align*} for every Borel set \(\eta\) of \(\mathbb{S}^{n-1}\). As it is the case with surface area measures, this concept can be extended to the notion of the \(L_p\) torsion measure of \(\Omega\). This is defined for every Borel set \(\eta\subseteq\mathbb{S}^{n-1}\) (and for \(p \in \mathbb{R}\)) as \[ \mu_p^{\mathrm{tor}}(\Omega,\eta) = \int_{\eta}h(\Omega,\theta)^{1-p}\,d\mu^{\mathrm{tor}}(\Omega,\theta). \] Of course, the case \(p = 1\) gives the usual torsion measure defined above. It is natural to extend the Mi…
reviewed by: Vitor Balestro
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 52A40
zbMATH DE Number: 7363708
zbMATH Keywords: torsional rigidity
zbMATH Keywords: \(L_p\)-Minkowski problem
zbMATH Keywords: polytope
MaRDI profile type: Publication
full work available at URL: j.aam.2021.102188
OpenAlex ID: W3138938155
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DOI: 10.1016/J.AAM.2021.102188
description: scientific article; zbMATH DE number 7363708