9 research outputs found
Analyticity of layer potentials and solvability of boundary value problems for divergence form elliptic equations with complex coefficients
We consider divergence form elliptic operators of the form L=-\dv
A(x)\nabla, defined in , ,
where the coefficient matrix is , uniformly
elliptic, complex and -independent. We show that for such operators,
boundedness and invertibility of the corresponding layer potential operators on
, is stable under
complex, perturbations of the coefficient matrix. Using a variant
of the Theorem, we also prove that the layer potentials are bounded and
invertible on whenever is real and symmetric (and
thus, by our stability result, also when is complex, is small enough and is real, symmetric,
and elliptic). In particular, we establish solvability of the Dirichlet and
Neumann (and Regularity) problems, with (resp. data, for
small complex perturbations of a real symmetric matrix. Previously,
solvability results for complex (or even real but non-symmetric) coefficients
were known to hold only for perturbations of constant matrices (and then only
for the Dirichlet problem), or in the special case that the coefficients
, , which corresponds to the Kato square
root problem