  The purpose of this article is to study extrapolation of solvability for
boundary value problems of elliptic systems in divergence form on the upper
half-space assuming De Giorgi type conditions. We develop a method allowing to
treat each boundary value problem independently of the others. We shall base
our study on solvability for energy solutions, estimates for boundary layers,
equivalence of certain boundary estimates with interior control so that
solvability reduces to a one-sided Rellich inequality. Our method then amounts
to extrapolating this Rellich inequality using atomic Hardy spaces,
interpolation and duality. In the way, we reprove the Regularity-Dirichlet
duality principle between dual systems and extend it to $H^1-BMO$. We also
exhibit and use a similar Neumann-Neumann duality principle.
