96 research outputs found
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^*\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.Comment: 51 page
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