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November  2010, 9(6): 1495-1505. doi: 10.3934/cpaa.2010.9.1495

Elastic Herglotz functions in the plane

J. A. Barceló 1, , M. Folch-Gabayet 2, , S. Pérez-Esteva 3, , A. Ruiz 4, and  M. C. Vilela 5,

1. 

ETSI de Caminos, Universidad Politécnica de Madrid, 28040 Madrid, Spain
2. 

Instituto de Matemáticas, Universidad Nacional Autónoma de MCiudad Universitaria, Ciudad Universitaria, México D.F., 04510, Mexico
3. 

Instituto de Matemáticas Unidad Cuernavaca, Universidad Nacional Autónoma de México, A.P. 273-3 ADMON 3, Cuernavaca, Mor., 62251, Mexico
4. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid, Spain
5. 

Departamento de Matemática Aplicada, Universidad de Valladolid,Plaza Santa Eulalia 9 y 11, 40005 Segovia, Spain

Received  October 2009 Revised  March 2010 Published  August 2010

We study spaces of solutions of the spectral Navier equation in the plane. We characterize the elastic Herglotz wave functions, namely the entire solutions $\mathbf{u}$ of the Navier equation with $L^2$ far-field-patterns. The characterization is in terms of a weighted $L^2$ norm involving $\mathbf{u}$ and its angular derivative $\partial_\theta \mathbf{u.}$ With respect to this norm, the space of elastic Herglotz wave functions is decomposed into the topological product of the compressional and shear elastic Herglotz wave functions. We also study the solutions of the Navier equation whose Lamé potentials are the Fourier transform of distributions in the circle. We prove that these are the entire solutions of the Navier equation with polynomial growth. This extends a result by Agmon for the Helmholtz equation.
Keywords: elastic Herglotz functions., Navier equation.
Mathematics Subject Classification: Primary: 35C15, 74B05; Secondary: 35J05, 46E1.
Citation: J. A. Barceló, M. Folch-Gabayet, S. Pérez-Esteva, A. Ruiz, M. C. Vilela. Elastic Herglotz functions in the plane. Communications on Pure and Applied Analysis, 2010, 9 (6) : 1495-1505. doi: 10.3934/cpaa.2010.9.1495
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