Cambridge Journal of Mathematics

Volume 9 (2021)

Number 1

Concavity properties of solutions to Robin problems

Pages: 177 – 212



Graziano Crasta (Dipartimento di Matematica, Sapienza Università di Roma, Italy)

Ilaria Fragalà (Dipartimento di Matematica, Politecnico di Milano, Italy)


We prove that the Robin ground state and the Robin torsion function are respectively $\operatorname{log}$-concave and $\frac{1}{2}$ -concave on an uniformly convex domain $\Omega \subset \mathbb{R}^N$ of class $\mathcal{C}^m$, with $[m - \frac{N}{2}] \geq 4$, provided the Robin parameter exceeds a critical threshold. Such threshold depends on $N, m$, and on the geometry of $\Omega$, precisely on the diameter and on the boundary curvatures up to order $m$.


Robin boundary conditions, eigenfunctions, torsion function, concavity

2010 Mathematics Subject Classification

35B65, 35E10, 35J15, 35J25

The authors have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

G. Crasta has been partially supported by Sapienza – Ateneo 2017 Project “Differential Models in Mathematical Physics” and Sapienza – Ateneo 2018 Project “Stationary and Evolutionary Problems in Mathematical Physics and Materials Science”.

Received 12 July 2020

Published 27 October 2021