The present paper commences the study of higher order differential equations in composition form. Specifically, we consider the equation Lu=divB*∇(adivA∇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 δ2 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 L2, and with optimal estimates in terms of nontangential maximal functions and square functions.
Bibliographical noteFunding Information:
Svitlana Mayboroda is partially supported by the Alfred P. Sloan Fellowship, the NSF CAREER Award DMS 1056004, and the NSF Materials Research Science and Engineering Center Seed Grant. Ariel Barton would like to thank Jill Pipher and Martin Dindos for helpful discussions concerning higher order differential equations, and Ana Grau de la Herran for many helpful discussions concerning the boundedness of linear operators that are far from being Calderón–Zygmund operators.
- Dirichlet problem
- Higher order elliptic equation