The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients

@article{Hofmann2013TheRP,
  title={The regularity problem for second order elliptic operators with complex-valued bounded measurable coefficients},
  author={Steve Hofmann and Carlos E. Kenig and Svitlana Mayboroda and Jill Pipher},
  journal={Mathematische Annalen},
  year={2013},
  volume={361},
  pages={863-907}
}
The present paper establishes a certain duality between the Dirichlet and Regularity problems for elliptic operators with $$t$$t-independent complex bounded measurable coefficients ($$t$$t being the transversal direction to the boundary). To be precise, we show that the Dirichlet boundary value problem is solvable in $$L^{p'}$$Lp′, subject to the square function and non-tangential maximal function estimates, if and only if the corresponding Regularity problem is solvable in $$L^p$$Lp. Moreover… 
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