Quasi-Exponential Solutions for Some PDE with Coefficients of Limited Regularity

  • Alexander Panchenko
Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 5)

Abstract

Let \(\Omega \subset {R^3}\) be a bounded Lipshitz domain, and let \(\zeta \in {C^3},\zeta \cdot \zeta = 0.\) In Ω, consider an elliptic equation
$$div\left( {a\nabla u} \right) + b\cdot \nabla u + cu = 0$$
with \(a \in {C^1}\left( {\bar \Omega } \right),b \in {L^\infty }\left( \Omega \right).\) Assume also that a is real valued and has a positive lower bound. We prove that for |ζ| sufficiently large, this equation has special quasi-exponential solutions of the form
$$u = {e^{ - \frac{1}{2}i\zeta \cdot x}}\left( {1 + w\left( {x,\zeta } \right)} \right)$$
depending on parameter ζ and such that \({\left\| \omega \right\|_{{L^2}\left( \Omega \right)}} = 0\left( {|\zeta {|^{ - \alpha }}} \right),\) for any α ∈ (0,1).

Keywords

Fundamental Solution Phase Function Convolution Operator Order Perturbation Tubular Neighborhood 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. BU] Brown R.M. and Uhlmann G. Uniqueness in the Inverse Conductivity Problem for Nonsmooth Conductivities in Two Dimensions. To appear in Comm. in PDE.Google Scholar
  2. [C]
    Calderon A.P. On an Inverse Boundary Value Problem. Seminar on Numerical Analysis and Its Applications to Continuum Physics, Rio de Janeiro, 1980, 65–73.Google Scholar
  3. ER] Eskin G. and Ralston J. Inverse Scattering Problem for the Scrödinger equation with Magnetic Potential at a Fixed Energy Comm.Math.Phys., 1995, (173), 173–199.Google Scholar
  4. [F]
    Fedoryuk M.V. Asymptotics: Integrals and Series, Nauka, Moscow, 1987. ( Russian).Google Scholar
  5. HeN] Henkin G.M. and Novikov R.G. The â-e q uation in the Multidimensional Inverse Scattering Problem Russian Math. Serveys, 1987,42, 101–180.Google Scholar
  6. H] Hörmander L. Analysis of Linear Partial Differential Operators,Springer-Verlag.Google Scholar
  7. I] Isakov V. Completeness of Products of Solutions and Some Inverse Problems for PDE. J Diff. Equations, 1991,92, 305–317.Google Scholar
  8. Il] Isakov V. Inverse Problems for PDE Springer-Verlag, 1997.Google Scholar
  9. N] Nachman A. Reconstruction from Boundary Measurements Ann. Math., 1988, 128, 531–577.Google Scholar
  10. NU] Nakamura G. and Uhlmann G. Global Uniqueness for an Inverse Boundary Value Problem Arising in Elasticity Invent. Math., 1994,118, 457474.Google Scholar
  11. S] Stein E.M. Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals Prinnceton University Press, 1993.Google Scholar
  12. Sh] Shubin M.A. Pseudodifferential Operators and Spectral Theory Springer-Verlag, 1987.Google Scholar
  13. SU] Sylvester J and Uhlmann G Global Uniqueness Theorem for an Inverse Boundary Value Problem Ann. Math., 1987,125, 153–169.Google Scholar
  14. Su] Sun Z. An Inverse Boundary Value Problem for Schrodinger Operator with Vector Potentials Trans. AMS, 1993, 338, 2, 953–969.Google Scholar
  15. T] Taylor M.E.Pseudeodifferential Operators and Nonlinear PDE Birkhauser, Boston, 1991.Google Scholar
  16. To] Tolmasky C.F. Exponentially Growing Solutions for Non-smooth First Order Perturbations of the Laplacian. To appear in SIAM J. Math. Anal.Google Scholar
  17. V] Vaiberg B.R. Asymptotic Methods in Equations of Mathematical Physics Gordon and Breach, 1989.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • Alexander Panchenko
    • 1
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA

Personalised recommendations