The Generalized Hölder and Morrey-Campanato Dirichlet Problems for Elliptic Systems in the Upper Half-Space

  • Juan José MarínEmail author
  • José María Martell
  • Marius Mitrea


We prove well-posedness results for the Dirichlet problem in \(\mathbb {R}^{n}_{+}\) for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces \(\mathscr{C}^{\omega }(\mathbb {R}^{n-1},\mathbb {C}^{M})\) and in generalized Morrey-Campanato spaces \(\mathscr{E}^{\omega ,p} (\mathbb {R}^{n-1},\mathbb {C}^{M})\) under certain assumptions on the growth function ω. We also identify a class of growth functions ω for which \(\mathscr{C}^{\omega }(\mathbb {R}^{n-1},\mathbb {C}^{M})=\mathscr{E}^{\omega ,p}(\mathbb {R}^{n-1},\mathbb {C}^{M})\) and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates.


Generalized Hölder space Generalized Morrey-Campanato space Dirichlet problem in the upper half-space Second order elliptic system Poisson kernel Lamé system Nontangential pointwise trace Fatou type theorem 

Mathematics Subject Classification (2010)

Primary: 35B65 35C15 35J47 35J57 35J67 42B37 Secondary: 35E99 42B35 


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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” (SEV-2015-0554). They also acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)” ERC agreement no. 615112 HAPDEGMT. The last author has been supported in part by the Simons Foundation grant #637481.


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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCMConsejo Superior de Investigaciones CientíficasMadridSpain
  2. 2.Department of MathematicsBaylor UniversityWacoUSA

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