  We consider divergence form elliptic operators of the form $L=-\dv
A(x)\nabla$, defined in $R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$,
where the $L^{\infty}$ 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
$L^2(\mathbb{R}^{n})=L^2(\partial\mathbb{R}_{+}^{n+1})$, is stable under
complex, $L^{\infty}$ 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, $\Vert
A-A^0\Vert_{\infty}$ is small enough and $A^0$ is real, symmetric, $L^{\infty}$
and elliptic). In particular, we establish solvability of the Dirichlet and
Neumann (and Regularity) problems, with $L^2$ (resp. $\dot{L}^2_1)$ 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
$A_{j,n+1}=0=A_{n+1,j}$, $1\leq j\leq n$, which corresponds to the Kato square
root problem.
