On the \(L_p\) torsional Minkowski problem for \(0 < p < 1\)
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 Minkowski problem for the \(L_p\) torsion measure: given a fixed number \(p\in\mathbb{R}\), what are the conditions on a Borel measure \(\mu\) on \(\mathbb{S}^{n-1}\) under which one can guarantee that \(\mu\) is the \(L_p\) torsion measure of some convex body \(\Omega \subseteq\mathbb{R}^n\)? According to the authors, this problem was previously investigated for the cases where \(p=1\) and \(p > 1\). The objective of the authors is to study the case where \(0 < p < 1\). The main theorem of the paper gives a sufficient condition for the case where \(0<p<1\). What the authors prove is the following: for \(0 < p < 1\), if \(\mu\) is a finite Borel measure on \(\mathbb{S}^{n-1}\) which is not concentrated on any closed hemisphere of \(\mathbb{S}^{n-1}\), then there exists a convex body \(\Omega\subseteq\mathbb{R}^n\) such that \(\mu = \mu_p^{\mathrm{tor}}(\Omega,\cdot)\).
scientific article; zbMATH DE number 7363708
scientific article