Concavity properties of solutions to Robin problems

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 $m$.

Keywords

Robin boundary conditions, eigenfunctions, torsion function, concavity

2010 Mathematics Subject Classification

35B65, 35E10, 35J15, 35J25

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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