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A Fatou Theorem and Poisson’s Integral Representation Formula for Elliptic Systems in the Upper Half-Space

  • Juan José MarínEmail author
  • José María Martell
  • Dorina Mitrea
  • Irina Mitrea
  • Marius Mitrea
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

Let L be a second-order, homogeneous, constant (complex) coefficient elliptic system in \({\mathbb {R}}^n\). The goal of this article is to prove a Fatou-type result, regarding the a.e. existence of the nontangential boundary limits of any null-solution u of L in the upper half-space, whose nontangential maximal function satisfies an integrability condition with respect to the weighted Lebesgue measure (1 + |x′|n−1)−1dx′ in \({\mathbb {R}}^{n-1}\equiv \partial {\mathbb {R}}^n_{+}\). This is the best result of its kind in the literature. In addition, we establish a naturally accompanying integral representation formula involving the Agmon-Douglis-Nirenberg Poisson kernel for the system L. Finally, we use this machinery to derive well-posedness results for the Dirichlet boundary value problem for L in \({\mathbb {R}}^n_{+}\) formulated in a manner which allows for the simultaneous treatment of a variety of function spaces.

Keywords

Fatou type theorem Poisson integral representation formula Nontangential maximal function Second-order elliptic system Dirichlet problem Hardy-Littlewood maximal operator Green function 

Mathematics Subject Classification (2010)

Primary 31A20 35C15 35J57 42B37; Secondary 42B25 

Notes

Acknowledgements

The first and second authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV2015-0554).

They also acknowledge support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT.

The third author has been supported in part by a Simons Foundation grant # 426669, the fourth author has been supported in part by Simons Foundation grants #318658 and #616050, while the fifth author has been supported in part by the Simons Foundation grant # 281566.

This work has been possible thanks to the support and hospitality of Temple University (USA), University of Missouri (USA), and ICMAT, Consejo Superior de Investigaciones Científicas (Spain). The authors express their gratitude to these institutions.

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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Juan José Marín
    • 1
    Email author
  • José María Martell
    • 1
  • Dorina Mitrea
    • 2
  • Irina Mitrea
    • 3
  • Marius Mitrea
    • 2
  1. 1.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCMConsejo Superior de Investigaciones CientíficasMadridSpain
  2. 2.Department of MathematicsBaylor UniversityWacoUSA
  3. 3.Department of MathematicsTemple UniversityPhiladelphiaUSA

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