Analyticity of layer potentials and L2 solvability of boundary value problems for divergence form elliptic equations with complex L∞ coefficients

We consider divergence form elliptic operators of the form L=−divA(x)∇, defined in Rn+1={(x,t)∈Rn×R}, n⩾2, where the L∞ coefficient matrix A is (n+1)×(n+1), uniformly elliptic, complex and t-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L2(Rn)=L2(∂R+n+1), is stable under complex, L∞ perturbations of the coefficient matrix. Using a variant of the Tb Theorem, we also prove that the layer potentials are bounded and invertible on L2(Rn) whenever A(x) is real and symmetric (and thus, by our stability result, also when A is complex, ‖A−A0‖∞ is small enough and A0 is real, symmetric, L∞ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L2 (resp. L˙12) data, for small complex perturbations of a real symmetric matrix. Previously, L2 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 Aj,n+1=0=An+1,j, 1⩽j⩽n, which corresponds to the Kato square root problem.