The Dirichlet problem for higher order equations in composition form

The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu = div B*del(a div A del 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. (C) 2013 Elsevier Inc. All rights reserved.