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<p>In this paper, we investigate a weighted overdetermined problem within the framework of Riemannian manifolds with density. Initially, by examining a Poisson problem associated with the drift Laplacian, we derive a Heintze-Karcher inequality and a soap bubble theorem that characterize geodesic balls in these spaces. Subsequently, by imposing both Dirichlet and Neumann boundary conditions, we establish a Serrin-type result in generalized cones, identifying metric balls as the unique solutions to this underlying overdetermined problem.</p>

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A Serrin’s type Problem in Weighted Manifolds: Soap Bubble Results and Rigidity in Generalized Cones

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Murilo Araújo,
  • Allan Freitas,
  • Márcio Santos

摘要

In this paper, we investigate a weighted overdetermined problem within the framework of Riemannian manifolds with density. Initially, by examining a Poisson problem associated with the drift Laplacian, we derive a Heintze-Karcher inequality and a soap bubble theorem that characterize geodesic balls in these spaces. Subsequently, by imposing both Dirichlet and Neumann boundary conditions, we establish a Serrin-type result in generalized cones, identifying metric balls as the unique solutions to this underlying overdetermined problem.