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<p>Using a variational formulation we consider the Dirichlet-to-Neumann operator on a connected open set Ω ⊂ ℝ<sup><i>d</i></sup> of finite volume, assuming only that the surface measure is locally finite on the boundary. Then the boundary may have infinite measure and trace properties become delicate. We show that this has consequences for the kernel of the Dirichlet-to-Neumann operator and characterise the situation in which a trace on Ω both exists and is unique.</p>

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The Dirichlet-to-Neumann operator on rough domains with finite volume

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  • 出版模式：

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  • 发表日期：

    2025年3月5日

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  • A. F. M. ter Elst,
  • Varun Narula

摘要

Using a variational formulation we consider the Dirichlet-to-Neumann operator on a connected open set Ω ⊂ ℝd of finite volume, assuming only that the surface measure is locally finite on the boundary. Then the boundary may have infinite measure and trace properties become delicate. We show that this has consequences for the kernel of the Dirichlet-to-Neumann operator and characterise the situation in which a trace on Ω both exists and is unique.