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

Abstract

We consider divergence form elliptic operators of the form $L=-\dv A(x)\nabla$, defined in $R^{n+1}=\{(x,t)\in R^n\times R\}$, $n\ge 2$, 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. ${\dot{L}}_1^2)$ 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.

References in corpus (3)

• The Green function estimates for strongly elliptic systems of second order
• A proof of the local $Tb$ Theorem for standard Calderón-Zygmund operators
• Lectures on the Kato square root problem

Cited by in corpus (35)

• The method of layer potentials in $L^p$ and endpoint spaces for elliptic operators with $L^{\infty }$ coefficients
• Global pointwise estimates for Green's matrix of second order elliptic systems
• The Dirichlet problem for elliptic systems with data in Köthe function spaces
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• Boundedness of single layer potentials associated to divergence form parabolic equations with complex coefficients
• Layer potentials for general linear elliptic systems
• A proof of the local $Tb$ Theorem for standard Calderón-Zygmund operators
• Extrapolation of the Dirichlet problem for elliptic equations with complex coefficients
• Boundary layers, Rellich estimates and extrapolation of solvability for elliptic systems
• Generalized local Tb Theorems for Square Functions, and applications
• On uniqueness results for Dirichlet problems of elliptic systems without DeGiorgi-Nash-Moser regularity
• Elliptic Partial Differential Equations with Complex Coefficients
• The $L^p$ Dirichlet boundary problem for second order Elliptic Systems with rough coefficients
• Square function/non-tangential maximal function estimates and the dirichlet problem for non-symmetric elliptic operators
• Boundary value problems in Lipschitz domains for equations with drifts
• Critical Perturbations for Second Order Elliptic Operators. Part I: Square function bounds for layer potentials
• Bounds on layer potentials with rough inputs for higher order elliptic equations
• Local Tb theorems and Hardy inequalities
• The $L^p$ Dirichlet and Regularity problems for second order Elliptic Systems with application to the Lamé system
• Solvability of elliptic systems with square integrable boundary data
• The Dirichlet problem for higher order equations in composition form
• Layer potentials beyond singular integral operators
• On domain of Poisson operators and factorization for divergence form elliptic operators
• Homogenization of Elliptic Boundary Value Problems in Lipschitz Domains
• Regularity theory for solutions to second order elliptic operators with complex coefficients and the $L^p$ Dirichlet problem
• The ${\dot{W}}^{-1,p}$ Neumann problem for higher order elliptic equations
• Higher-order elliptic equations in non-smooth domains: history and recent results
• Extrapolation for the $L^p$ Dirichlet Problem in Lipschitz domains
• Remark on the Helmholtz decomposition in domains with noncompact boundary
• A new approach to solvability of some elliptic pde's with square integrable boundary data
• The $L^p$ Neumann problem for higher order elliptic equations
• The Neumann problem for higher order elliptic equations with symmetric coefficients
• On Poisson operators and Dirichlet-Neumann maps in H^s for divergence form elliptic operators with Lipschitz coefficients
• The Regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients
• $L^2$ estimates for commutators of the Dirichlet-to-Neumann Map associated to elliptic operators with complex-valued bounded measurable coefficients on ${\mathbb{R}}_{+}^{n+1}$