Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems |
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Authors: | Pascal Auscher Steve Hofmann |
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Institution: | a Université de Paris-Sud, UMR du CNRS 8628, 91405 Orsay Cedex, France b Matematiska institutionen, Stockholms universitet, 106 91 Stockholm, Sweden c Mathematics Department, University of Missouri, Columbia, MO 65211, USA |
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Abstract: | We prove that Neumann, Dirichlet and regularity problems for divergence form elliptic equations in the half-space are well posed in L2 for small complex L∞ perturbations of a coefficient matrix which is either real symmetric, of block form or constant. All matrices are assumed to be independent of the transversal coordinate. We solve the Neumann, Dirichlet and regularity problems through a new boundary operator method which makes use of operators in the functional calculus of an underlaying first order Dirac type operator. We establish quadratic estimates for this Dirac operator, which implies that the associated Hardy projection operators are bounded and depend continuously on the coefficient matrix. We also prove that certain transmission problems for k-forms are well posed for small perturbations of block matrices. |
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Keywords: | Neumann problem Dirichlet problem Elliptic equation Non-symmetric coefficients Dirac operator Functional calculus Quadratic estimates Perturbation theory Carleson measure |
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