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2010, Volume 9, Issue 6: 1495-1505. Doi: 10.3934/cpaa.2010.9.1495
This issue Previous Article Global in time solution and time-periodicity for a smectic-A liquid crystal model Next Article Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case

Elastic Herglotz functions in the plane

  • 1.

    ETSI de Caminos, Universidad Politécnica de Madrid, 28040 Madrid

  • 2.

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

  • 3.

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

  • 4.

    Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049, Madrid

  • 5.

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

Received:  October 2009
Revised:  March 2010
Published:  November 2010
Abstract Related Papers Cited by
    Abstract
  • 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.
    Mathematics Subject Classification: Primary: 35C15, 74B05; Secondary: 35J05, 46E15.

    Citation:
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