Analyticity of layer potentials and $L^2$ solvability of boundary value problems for divergence form elliptic equations with complex $L^{\mathrm{\infty }}$ coefficients

We consider divergence form elliptic operators of the form $L=-\dv A(x)\nabla$,definedin$R^{n+1} = {(x,t)\in R^n \times R }$,$n \geq 2$, where the $L^{\mathrm{\infty }}$ coefficient matrix $A$ is $(n+1)\times (n+1)$, uniformly elliptic, complex and $t$-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on $L^2({\mathbb{R}}^n)=L^2(\mathrm{\partial }{\mathbb{R}}_{+}^{n+1})$, is stable under complex, $L^{\mathrm{\infty }}$ perturbations of the coefficient matrix. Using a variant of the $Tb$ Theorem, we also prove that the layer potentials are bounded and invertible on $L^2({\mathbb{R}}^n)$ whenever $A(x)$ is real and symmetric (and thus, by our stability result, also when $A$ is complex, $\Vert A-A^0\Vert_{\infty}$issmallenoughand$A^0$isreal,symmetric,$L^{\infty}$ and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with $L^2$ (resp. ${\dot{L}}_1^2)$ data, for small complex perturbations of a real symmetric matrix. Previously, $L^2$ 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 $A_{j,n+1}=0=A_{n+1,j}$, $1\le j\le n$, which corresponds to the Kato square root problem.