Concavity Properties of Robin Solutions on $C^{3,1}$ Uniformly Convex Domains

RESUMO

We prove that, on a bounded uniformly convex domain of class $C^{3,1}$, the first Robin eigenfunction is strictly log-concave and the Robin torsion function is strictly $1/2$-concave for all sufficiently large Robin parameters. This means that Conjecture 1.1 of Andrews, Clutterbuck and Hauer holds for uniformly convex $C^{3,1}$ domains in any dimension, and settles an open problem posed by Crasta and Fragalà when the domain has $C^{3,1}$ regularity. Our proof derives suitable uniform $C^2$-decay estimates by maximum principle arguments, thereby removing the imposed higher-order boundary regularity assumption required previously.