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We consider divergence form elliptic operators of the form $L=-\dvA(x)\nabla$, defined in $R^{n+1} = \{(x,t)\in R^n \times R \}$, $n \geq 2$,where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformlyelliptic, 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 undercomplex, $L^{\infty}$ perturbations of the coefficient matrix. Using a variantof the $Tb$ Theorem, we also prove that the layer potentials are bounded andinvertible on $L^2(\mathbb{R}^n)$ whenever $A(x)$ is real and symmetric (andthus, by our stability result, also when $A$ is complex, $\VertA-A^0\Vert_{\infty}$ is small enough and $A^0$ is real, symmetric, $L^{\infty}$and elliptic). In particular, we establish solvability of the Dirichlet andNeumann (and Regularity) problems, with $L^2$ (resp. $\dot{L}^2_1)$ data, forsmall complex perturbations of a real symmetric matrix. Previously, $L^2$solvability results for complex (or even real but non-symmetric) coefficientswere known to hold only for perturbations of constant matrices (and then onlyfor 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 squareroot problem.

advances in mathematics

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