Advertisement

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
Article
  • 2 Downloads

Abstract

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.

Keywords

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 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgments

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.

References

  1. 1.
    Bennett, C., Sharpley, R.: Interpolation of Operators Pure and Applied Mathematics, vol. 129. Academic Press Inc, Boston (1988)Google Scholar
  2. 2.
    Fiorenza, A., Krbec, M.: Indices of Orlicz spaces and some applications. Comment Math. Univ. Carolin. 38(3), 433–451 (1997)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Hofmann, S., Martell, J.M., Mayboroda, S.: Transference of scale-invariant estimates from Lipschitz to non-tangentially accessible to uniformly rectifiable domains, arXiv:1904.13116
  4. 4.
    Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344(1), 37–116 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Lerner, A.K., Pérez, C.: Self-improving properties of generalized Poincaré type inequalities through rearrangements. Math. Scand. 97(2), 217–234 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Martell, J.M., Mitrea, D., Mitrea, I., Mitrea, M.: The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO. Anal. PDE. 12(3), 605–720 (2019)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Martell, J.M., Mitrea, D., Mitrea, I., Mitrea, M.: The Dirichlet problem for elliptic systems with data in Köthe function spaces. Rev. Mat. Iberoam. 32(3), 913–970 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Maz’ya, V., Mitrea, M., Shaposhnikova, T.: The Dirichlet problem in Lipschitz domains with boundary data in Besov spaces for higher order elliptic systems with rough coe<cients. J. d’Analyse Mathématique 110(1), 167–239 (2010)CrossRefzbMATHGoogle Scholar
  9. 9.
    Meyers, N.G.: Mean oscillation over cubes and Hölder continuity. Proc. Amer. Math. Soc. 15(5), 717–721 (1964)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Mitrea, D.: Distributions, Partial Differential Equations, and Harmonic Analysis, Universitext. Springer, New York (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Neri, U.: Some properties of functions with bounded mean oscillation. Studia Math. 61(1), 63–75 (1977). MR 0445210MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pérez, C.: Calderón-Zygmund theory related to Poincaré-Sobolev inequalities, fractional integrals and singular integral operators. In: Lukes, J., Pick, L. (eds.) Function Spaces, Nonlinear Analysis and Applications. Lecture notes of the Spring Lectures in Analysis, pp. 31–94. Charles University and Academy of Sciences (1999). Available at http://grupo.us.es/anaresba/trabajos/carlosperez/31.pdf

Copyright information

© 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

Personalised recommendations