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arXiv · 2609.06223

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

Dong Ye·Dekai Zhang
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Abstract

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 Fragal\`a 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.

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Dong Ye, Dekai Zhang. 2026-09-05. Concavity Properties of Robin Solutions on $C^{3,1}$ Uniformly Convex Domains. https://arxiv.org/abs/2609.06223

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