Concavity properties of solutions to Robin problems (Q2041481)

The authors of this article consider two problems for an open uniformly convex bounded domain \(\Omega\) belonging to class \(\mathcal{C}^m\) in \(N\) dimensions with a boundary \(\partial \Omega\) satisfying \(\lfloor m-N/2\rfloor \geq 4\). Specifically, the Robin ground state of \(\Omega\) is considered: \(-\Delta u=\lambda^\beta u\) in \(\Omega\) and \(\partial_\nu u+\beta u=0\) on \(\partial \Omega\) where \(\lambda^\beta\) is the first Robin eigenvalue of \(\Omega\). Moreover, the Robin torsion function of \(\Omega\) is considered: \(-\Delta u=1\) in \(\Omega\) and \(\partial_\nu u+\beta u=0\) on \(\partial \Omega\). It is shown that they are \(\log\)-concave and \(1/2\)-concave, respectively under the assumption that the Robin parameter is greater a critical threshold depending on \(N\), \(m\) as well as on the geometry of the domain; that is, on the diameter and the curvature.