Geodesic
A class of solutions of the n-dimensional generalized Helmholtz equation which describes generalized Weingarten hypersurfaces
Serdica Mathematical Journal, Tome 50 (2024) no. 1, pp. 1-34 Cet article a éte moissonné depuis la source Bulgarian Digital Mathematics Library

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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.
Keywords: Helmholtz equation, generalized Helmholtz equation, Weingarten hypersurfaces, biharmonic functions, support function, 35J05, 53C42 
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     title = {A class of solutions of the n-dimensional generalized {Helmholtz} equation which describes generalized {Weingarten} hypersurfaces},
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Corro, Armando; Riveros, Carlos; Carretero, José. A class of solutions of the n-dimensional generalized Helmholtz equation which describes generalized Weingarten hypersurfaces. Serdica Mathematical Journal, Tome 50 (2024) no. 1, pp. 1-34. http://geodesic.mathdoc.fr/item/SMJ2_2024_50_1_a0/