American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Positive solutions and bifurcation phenomena for nonlinear elliptic equations of logistic type: The superdiffusive case
  • CPAA Home
  • This Issue
  • Next Article
    Global in time solution and time-periodicity for a smectic-A liquid crystal model
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 & Applied Analysis, 2010, 9 (6) : 1495-1505. doi: 10.3934/cpaa.2010.9.1495
[1]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. I: Numerical tests and examples. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 41-74. doi: 10.3934/dcdsb.2010.14.41
[2]

Gerard Gómez, Josep–Maria Mondelo, Carles Simó. A collocation method for the numerical Fourier analysis of quasi-periodic functions. II: Analytical error estimates. Discrete & Continuous Dynamical Systems - B, 2010, 14 (1) : 75-109. doi: 10.3934/dcdsb.2010.14.75
[3]

Ricardo Almeida, Agnieszka B. Malinowska. Fractional variational principle of Herglotz. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2367-2381. doi: 10.3934/dcdsb.2014.19.2367
[4]

Seongyeon Kim, Yehyun Kwon, Ihyeok Seo. Strichartz estimates and local regularity for the elastic wave equation with singular potentials. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1897-1911. doi: 10.3934/dcds.2020344
[5]

Dina Tavares, Ricardo Almeida, Delfim F. M. Torres. Fractional Herglotz variational problems of variable order. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 143-154. doi: 10.3934/dcdss.2018009
[6]

Ezzeddine Zahrouni. On the Lyapunov functions for the solutions of the generalized Burgers equation. Communications on Pure & Applied Analysis, 2003, 2 (3) : 391-410. doi: 10.3934/cpaa.2003.2.391
[7]

Kuijie Li, Tohru Ozawa, Baoxiang Wang. Dynamical behavior for the solutions of the Navier-Stokes equation. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1511-1560. doi: 10.3934/cpaa.2018073
[8]

C. Foias, M. S Jolly, I. Kukavica, E. S. Titi. The Lorenz equation as a metaphor for the Navier-Stokes equations. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 403-429. doi: 10.3934/dcds.2001.7.403
[9]

Stéphane Mischler, Clément Mouhot. Stability, convergence to the steady state and elastic limit for the Boltzmann equation for diffusively excited granular media. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 159-185. doi: 10.3934/dcds.2009.24.159
[10]

Shuya Kanagawa, Ben T. Nohara. The nonlinear Schrödinger equation created by the vibrations of an elastic plate and its dimensional expansion. Conference Publications, 2013, 2013 (special) : 415-426. doi: 10.3934/proc.2013.2013.415
[11]

Rainer Brunnhuber, Barbara Kaltenbacher, Petronela Radu. Relaxation of regularity for the Westervelt equation by nonlinear damping with applications in acoustic-acoustic and elastic-acoustic coupling. Evolution Equations & Control Theory, 2014, 3 (4) : 595-626. doi: 10.3934/eect.2014.3.595
[12]

Irena Lasiecka, Roberto Triggiani. A sharp trace result on a thermo-elastic plate equation with coupled hinged/Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 1999, 5 (3) : 585-598. doi: 10.3934/dcds.1999.5.585
[13]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[14]

Daniel Alpay, Eduard Tsekanovskiĭ. Subclasses of Herglotz-Nevanlinna matrix-valued functtons and linear systems. Conference Publications, 2001, 2001 (Special) : 1-13. doi: 10.3934/proc.2001.2001.1
[15]

I. Moise, Roger Temam. Renormalization group method: Application to Navier-Stokes equation. Discrete & Continuous Dynamical Systems, 2000, 6 (1) : 191-210. doi: 10.3934/dcds.2000.6.191
[16]

Igor Kukavica, Mohammed Ziane. Regularity of the Navier-Stokes equation in a thin periodic domain with large data. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 67-86. doi: 10.3934/dcds.2006.16.67
[17]

Mohamad Rachid. Incompressible Navier-Stokes-Fourier limit from the Landau equation. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021017
[18]

Linglong Du. Long time behavior for the visco-elastic damped wave equation in $\mathbb{R}^n_+$ and the boundary effect. Networks & Heterogeneous Media, 2018, 13 (4) : 549-565. doi: 10.3934/nhm.2018025
[19]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete & Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006
[20]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4593-4610. doi: 10.3934/dcds.2015.35.4593

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (47)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences