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)εRⁿ×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 A_j,n+1=0=A_n+1,j, 1≤j≤n, which corresponds to the Kato square root problem.