9 research outputs found

    Analyticity of layer potentials and L2L^{2} solvability of boundary value problems for divergence form elliptic equations with complex L∞L^{\infty} coefficients

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    We consider divergence form elliptic operators of the form L=-\dv A(x)\nabla, defined in Rn+1={(x,t)∈Rn×R}R^{n+1} = \{(x,t)\in R^n \times R \}, n≥2n \geq 2, where the L∞L^{\infty} coefficient matrix AA is (n+1)×(n+1)(n+1)\times(n+1), uniformly elliptic, complex and tt-independent. We show that for such operators, boundedness and invertibility of the corresponding layer potential operators on L2(Rn)=L2(∂R+n+1)L^2(\mathbb{R}^{n})=L^2(\partial\mathbb{R}_{+}^{n+1}), is stable under complex, L∞L^{\infty} perturbations of the coefficient matrix. Using a variant of the TbTb Theorem, we also prove that the layer potentials are bounded and invertible on L2(Rn)L^2(\mathbb{R}^n) whenever A(x)A(x) is real and symmetric (and thus, by our stability result, also when AA is complex, ∥A−A0∥∞\Vert A-A^0\Vert_{\infty} is small enough and A0A^0 is real, symmetric, L∞L^{\infty} and elliptic). In particular, we establish solvability of the Dirichlet and Neumann (and Regularity) problems, with L2L^2 (resp. L˙12)\dot{L}^2_1) data, for small complex perturbations of a real symmetric matrix. Previously, L2L^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 Aj,n+1=0=An+1,jA_{j,n+1}=0=A_{n+1,j}, 1≤j≤n1\leq j\leq n, which corresponds to the Kato square root problem
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