Elliptic Partial Differential Equations with Almost-Real Coefficients
Ariel Barton · Memoirs of the American Mathematical Society · 2012
In this monograph we investigate 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. We show that for such operators, the Dirichlet problem with boundary data in L q L^q can be solved for q > ∞ q>\infty large enough. We also show that the Neumann and regularity problems with boundary data in L p L^p can be solved for p > 1 p>1 small enough, and provide an endpoint result at p = 1 p=1 .