On the strict concavity of the harmonic radius in dimension \(N\geq 3\)
The paper provides a result on the strict concavity of the harmonic radius of a convex domain in \({\mathbb R}^N\) for \(N\geq 3.\) Precisely, given a domain \(\Omega\subset {\mathbb R}^N\) and denote by \(G_x(\cdot)\) the Green function for the Laplacian in \(\Omega.\) Indeed, \(G_x\) is decomposable into \[ G_x(\cdot)=F_x(\cdot)-H_x(\cdot) \] with singular part \(F_x\) given by the fundamental solution of the Laplacian (\(\forall y\in\Omega\) \(F_x(y)=F(|x-y|)=c_N |x-y|^{2-N},\) \(c_N=\text{const}\)) and \(H_x(\cdot)\) is a harmonic function in \(\Omega\) attaining the values of \(F_x(\cdot)\) on \(\partial\Omega.\) The Robin function \(t(\cdot)\colon\;\Omega\to {\mathbb R}\) is then given by \[ t(x)=H_x(x)\quad \forall x\in \Omega, \] the harmonic radius is \[ r(x)=F^{-1}(t(x))\quad \forall x\in \Omega \] and the harmonic center of \(\Omega\) is a point where \(r(\cdot)\) attains its maximum/minimum. (In the two-dimensional case \(r(\cdot)\) is known as conformal radius.) These notions play a main role when studying elliptic boundary value problems involving concentration of energy and the question of uniqueness of the harmonic center is crucial therein. Employing refined results on capacity theory (Borell's inequality, Brunn-Minkowski inequality, properties of capacity potentials, etc.), the authors establish strict concavity of the harmonic radius when \(N\geq 3\) unless \(\Omega\) is a cylinder, a cone or the translate of a cone. That result implies in particular that the harmonic center of a convex bounded set is unique and the Robin function is strictly convex.
scientific article; zbMATH DE number 1770855
scientific article