  The present paper commences the study of higher order differential equations
in composition form. Specifically, we consider the equation Lu=\Div
B^*\nabla(a\Div A\nabla u)=0, where A and B are elliptic matrices with
complex-valued bounded measurable coefficients and a is an accretive function.
Elliptic operators of this type naturally arise, for instance, via a pull-back
of the bilaplacian \Delta^2 from a Lipschitz domain to the upper half-space.
More generally, this form is preserved under a Lipschitz change of variables,
contrary to the case of divergence-form fourth order differential equations. We
establish well-posedness of the Dirichlet problem for the equation Lu=0, with
boundary data in L^2, and with optimal estimates in terms of nontangential
maximal functions and square functions.
