The Dirichlet problem for higher order equations in composition form

TL;DR

This paper studies higher order elliptic equations in composition form, establishing well-posedness of the Dirichlet problem with boundary data in L^2 and providing optimal estimates.

Contribution

It introduces a new class of higher order differential equations in composition form and proves well-posedness and estimates for the Dirichlet problem in this setting.

Findings

Well-posedness of the Dirichlet problem for the composition form equation.

Optimal estimates in terms of nontangential maximal functions and square functions.

Applicability to equations arising from Lipschitz domain transformations.

Abstract

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.