# American Institute of Mathematical Sciences

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

## Elastic Herglotz functions in the plane

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.
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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