  In a paper from 1996, D. Jerison and C. Kenig among other results provided a
$H^{1/2}$ regularity result for the Dirichlet problem for the Laplace equation
in Lipschitz domains. In this article, we adopt a Hilbertian approach to
construct two Riesz bases for $L^2(\partial \Omega),$ which will allow to find
in a different way some of the results of D. Jerison and C. Kenig, and G.
Savar\'{e} (1998) about the regularity issue of the Laplace equation.
