  For elliptic systems with block structure in the upper half-space and
t-independent coefficients, we settle the study of boundary value problems by
proving compatible well-posedness of Dirichlet, regularity and Neumann problems
in optimal ranges of exponents. Prior to this work, only the two-dimensional
situation was fully understood. In higher dimensions, partial results for
existence in smaller ranges of exponents and for a subclass of such systems had
been established. The presented uniqueness results are completely new. We also
elucidate optimal ranges for problems with fractional regularity data. Methods
use and improve, with some new results, all the machinery developed over the
last two decades to study such problems: the Kato square root estimates and
Riesz transforms, Hardy spaces associated to operators, off-diagonal estimates,
non-tangential estimates and square functions and abstract layer potentials to
replace fundamental solutions in the absence of local regularity of solutions.
This self-contained monograph provides a comprehensive overview on the field
and unifies many earlier results that have been obtained by a variety of
methods.
