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<p>The eigenfunction problem of the Laplace operator is studied in the angular domain with the Robin-type boundary condition on the upper side of the angle and the Neumann-type boundary condition on the bottom side of the angle. From physical point of view, such eigenfunctions describe waves over sloping beach. Negative values of the spectral parameter are considered. The eigenfunction of the essential spectrum is obtained and a special case of eigenfunction, which is elementary function, is studied. The Sommerfeld integral representation of the eigenfunction of the negative part of the essential spectrum of the Laplace operator is obtained. Moreover, its asymptotic is calculated far away from the angle vertex. It is bounded on the top side of the angle and vanishes exponentially in the angle interior with its bottom side. Thus the eigenfunction of essential spectrum behaves like a surface wave. Bibliography: 11 titles.</p>

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EIGENFUNCTIONS OF ESSENTIAL SPECTRUM FOR THE LAPLACE OPERATOR IN ANGLE WITH ROBIN-NEUMANN BOUNDARY CONDITIONS

  • 期刊论文

  • 发表日期：

    2025年9月1日

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  • M. A. Lyalinov,
  • N. S. Fedorov

摘要

The eigenfunction problem of the Laplace operator is studied in the angular domain with the Robin-type boundary condition on the upper side of the angle and the Neumann-type boundary condition on the bottom side of the angle. From physical point of view, such eigenfunctions describe waves over sloping beach. Negative values of the spectral parameter are considered. The eigenfunction of the essential spectrum is obtained and a special case of eigenfunction, which is elementary function, is studied. The Sommerfeld integral representation of the eigenfunction of the negative part of the essential spectrum of the Laplace operator is obtained. Moreover, its asymptotic is calculated far away from the angle vertex. It is bounded on the top side of the angle and vanishes exponentially in the angle interior with its bottom side. Thus the eigenfunction of essential spectrum behaves like a surface wave. Bibliography: 11 titles.