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In this work we study the Helmholtz equation with boundary condition in the exterior of a ball of radius \(\varepsilon \) in dimension 3, in the radial case. The boundary condition is the Robin condition with a power type weight. Taking the limit \(\varepsilon \rightarrow 0\) , we show that the solution of the problem doesn’t feel the initial hole, unless if the power of the weight is 1. In this case a singular perturbation appears and the resolvent behaves exactly like the resolvent of \(\Delta _\alpha \) , the Laplace operator with point-like interaction.

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Resolvent Limits of Exterior Boundary Value Problems and Singular Perturbation of Laplace Operator in 3D

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  • 发表日期：

    2025年

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  • Vladimir Georgiev,
  • Mario Rastrelli

摘要

In this work we study the Helmholtz equation with boundary condition in the exterior of a ball of radius \(\varepsilon \) in dimension 3, in the radial case. The boundary condition is the Robin condition with a power type weight. Taking the limit \(\varepsilon \rightarrow 0\) , we show that the solution of the problem doesn’t feel the initial hole, unless if the power of the weight is 1. In this case a singular perturbation appears and the resolvent behaves exactly like the resolvent of \(\Delta _\alpha \) , the Laplace operator with point-like interaction.