Uniform meyer solution to the three dimensional cauchy problem for laplace equation

Abstract

We consider the three dimensional Cauchy problem for the Laplace equation

$\left\{ \begin{gathered} u_{xx} (x,y,z) + u_{yy} (x,y,z) + u_{zz} (x,y,z) = 0,x \in R,y \in R,0 < z \leqslant 1, \hfill \\ u(x,y,0) = g(x,y),x \in R,y \in R, \hfill \\ u_z (x,y,0) = 0,x \in R,y \in R, \hfill \\ \end{gathered} \right. $

where the data is given at z = 0 and a solution is sought in the region x,y ∈ R,0 < z < 1. The problem is ill-posed, the solution (if it exists) doesn’t depend continuously on the initial data. Using Galerkin method and Meyer wavelets, we get the uniform stable wavelet approximate solution. Furthermore, we shall give a recipe for choosing the coarse level resolution.