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<p>The purpose of this paper is to establish a quantitative version of the Talenti comparison principle for solutions to the Poisson equation with Robin boundary conditions. This quantitative enhancement is proved in terms of the asymmetry of the domain. The key role is played by a careful analysis of the propagation of asymmetry for the level sets of the solutions to a PDE. As a byproduct, we obtain an alternative proof of the quantitative Saint-Venant inequality for the Robin torsion and, in the planar case, of the quantitative Faber-Krahn inequality for the first Robin eigenvalue. In addition, we complete the framework of the rigidity result of the Talenti inequalities with Robin boundary conditions.</p>

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A quantitative Talenti-type comparison result with Robin boundary conditions

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年8月1日

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  • Vincenzo Amato,
  • Rosa Barbato,
  • Simone Cito,
  • Alba Lia Masiello,
  • Gloria Paoli

摘要

The purpose of this paper is to establish a quantitative version of the Talenti comparison principle for solutions to the Poisson equation with Robin boundary conditions. This quantitative enhancement is proved in terms of the asymmetry of the domain. The key role is played by a careful analysis of the propagation of asymmetry for the level sets of the solutions to a PDE. As a byproduct, we obtain an alternative proof of the quantitative Saint-Venant inequality for the Robin torsion and, in the planar case, of the quantitative Faber-Krahn inequality for the first Robin eigenvalue. In addition, we complete the framework of the rigidity result of the Talenti inequalities with Robin boundary conditions.