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Vol. 50 No. 1 (2024)
Serdica Mathematical Journal.

Articles

A class of solutions of the n-dimensional generalized Helmholtz equation which describes generalized Weingarten hypersurfaces

https://doi.org/10.55630/serdica.2024.50.1-34

Published 05-04-2024

  • Armando M. V. Corro
    • ,
  • Carlos M. C. Riveros
    • ,
  • José L. T. Carretero

Armando M. V. Corro
Instituto de Matemática e Estatística, Universidade Federal de Goiás

Carlos M. C. Riveros
Departamento de Matemática, Universidade de Brasília

José L. T. Carretero
Departamento de Matemática, Universidade de Brasília

Abstract

In this paper, we introduce the \(n\)-dimensional generalized Helmholtz equation and present explicit solutions to this equation in terms of biharmonic functions, in particular, we get solutions that depend on holomorphic functions. Also, we present explicit radial solutions for this equation and we provide explicit solutions to the \(n\)-dimensional Helmholtz equation. In addition, as an application we introduced two classes of generalized Weingarten hypersurfaces, namely, the RSHGW-hypersurfaces and the RSGW-hypersurfaces, associated with solutions of the \(n\)-dimensional generalized Helmholtz equation and classify the RSHGW-hypersurfaces of rotation. For \(n=2\), we obtain a Weierstrass type representation for these surfaces which depend of three holomorphic functions and we classify the RSHGW-surfaces and the RSGW-surfaces of rotation.

2020 Mathematics Subject Classification:

35J05, 53C42

Keywords

Helmholtz equation, generalized Helmholtz equation, Weingarten hypersurfaces, biharmonic functions, support function

PDF

Author Details

Armando M. V. Corro

Instituto de Matemática e Estatística
Universidade Federal de Goiás
CEP 74001-970, Goiânia, GO, Brazil
e-mail: avcorro@gmail.com

Carlos M. C. Riveros

Departamento de Matemática

Universidade de Brasília

CEP 70910-900, Brasília, DF, Brazil

e-mail: carlos@mat.unb.br

José L. T. Carretero

Departamento de Matemática

Universidade de Brasília

CEP 70910-900, Brasília, DF, Brazil

e-mail: jlteruel@mat.unb.br

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