@Article{ATA-27-265,
author = {Jinru Wang and Weifang Wang},
title = {Uniform Meyer Solution to the Three Dimensional Cauchy Problem for Laplace Equation},
journal = {Analysis in Theory and Applications},
year = {2011},
volume = {27},
number = {3},
pages = {265--277},
abstract = {<p style="text-align: justify;">We consider the three dimensional Cauchy problem for the Laplace equation$$\left\{\begin{array}{ll}u_{xx}(x,y, z)+u_{yy}(x,y, z)+u_{zz}(x,y, z) = 0, &nbsp;&amp; x \in R, y \in R, 0 &lt; z \leq 1,\\u(x,y,0) = g(x,y), &amp; x \in R, y \in R,\\u_z(x,y,0) = 0, &nbsp;&amp; &nbsp;x \in R, y \in R,\end{array}\right.$$where the data is given at $z = 0$ and a solution is sought in the region $x,y \in R$, $0 &lt; z &lt; 1$. 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.</p>},
issn = {1573-8175},
doi = {https://doi.org/10.1007/s10496-011-0265-6},
url = {http://global-sci.org/intro/article_detail/ata/4599.html}
}