Elliptic Partial Differential Equations with Almost-Real Coefficients

In this monograph the author investigates divergence-form elliptic partial differential equations in two-dimensional Lipschitz domains whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates.

He shows that for such operators, the Dirichlet problem with boundary data in Lq can be solved for q<∞ large enough. He also shows that the Neumann and regularity problems with boundary data in Lp can be solved for p>1 small enough, and provide an endpoint result at p=1.