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)


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).


Fundamental Solution Phase Function Convolution Operator Order Perturbation Tubular Neighborhood 
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© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

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

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