The Brunn-Minkowski inequality for the \(n\)-dimensional logarithmic capacity of convex bodies
Let \({\mathcal M}(u, r)\) denote the mean value of a nonnegative Borel measurable function \(u\) over the sphere \(\{|x|= r\}\) in \(\mathbb{R}^n\) \((n\geq 2)\) and let \({\mathcal M}_p(u,r)= \{{\mathcal M}(|u|^p, r)\}^{1/p}\). Also, let \(G_2(x,y)\) denote the Green function for the biharmonic operator \(\Delta^2\) on the unit ball \(B\) and, for any (nonnegative) measure \(\mu\) on \(B\), let \(G_2\mu\) denote the biharmonic Green potential \(\int_B G_2(\cdot,y)\,d\mu(y)\). This paper shows that \(\lim_{r\to 1}(1- r)^{n-2-(n-1)/p}{\mathcal M}_p(G_2\mu, r)= 0\) if \(1\leq p< (n-1)/(n-4)\) in the case \(n\geq 5\), and \(1\leq p< \infty\) in the case \(n\leq 4\). If \(n\geq 5\) and \((n-1)/(n-4)\leq p< (n-1)/(n- 5)\), then one can still assert that the lower limit of the above expression is \(0\). The corresponding results for the Laplace operator are due to the reviewer [Proc. Am. Math. Soc. 103, No. 3, 861--869 (1988; Zbl 0672.31005)]. The authors also characterize biharmonic Green potentials, among all ``superbiharmonic'' functions \(u\) on \(B\) (corresponding to the distributional inequality \(\Delta^2u\geq 0\)) by the condition \(\liminf_{r\to1}(1- r)^{-1}{\mathcal M}_1(u, r)= 0\).
scientific article; zbMATH DE number 2162297