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

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).