Skip to main content
SpringerLink
Log in

Boundary regularity for the Poisson equation in reifenberg-flat domains

  • Conference paper
Geometric Partial Differential Equations proceedings

Part of the book series: CRM Series ((CRMSNS,volume 15))

  • 1225 Accesses

Abstract

This paper is devoted to the investigation of the boundary regularity for the Poisson equation

$$ \left\{ {\begin{array}{*{20}c} { - \Delta u = f\quad in\Omega } \\ {u = 0\quad on\partial \Omega } \\ \end{array} } \right. $$

where f belongs to some L p (Ω) and Ω is a Reifenberg-flat domain of ℝN. More precisely, we prove that given an exponent α ∈ (0,1), there exists an ε > 0 such that the solution u to the previous system is locally Hölder continuous provided that Ω is (ε, r 0)-Reifenberg-flat. The proof is based on Alt-Caffarelli-Friedman’s monotonicity formula and Morrey-Campanato theorem.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 24.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 34.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. W. Alt, L. A. Caffarelli and A. Friedman, Variational problems with two phases and their free boundaries, Trans. Amer. Math. Soc. (2) 282 (1984), 431–461.

    Article  MATH  MathSciNet  Google Scholar 

  2. S. Byun and L. Wang, Elliptic equations with BMO nonlinearity in Reifenberg domains, Adv. Math. (6) 219 (2008), 1937–1971.

    Article  MATH  MathSciNet  Google Scholar 

  3. S. Byun and L. Wang, Gradient estimates for elliptic systems in non-smooth domains, Math. Ann. (3) 341 (2008), 629–650.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. Byun, L. Wang and S. Zhou, Nonlinear elliptic equations with BMO coefficients in Reifenberg domains, J. Funct. Anal. (1) 250 (2007), 167–196.

    Article  MATH  MathSciNet  Google Scholar 

  5. S.-S. Byun and L. Wang, Fourth-order parabolic equations with weak BMO coefficients in Reifenberg domains, J. Differential Equations (11) 245 (2008), 3217–3252.

    Article  MATH  MathSciNet  Google Scholar 

  6. S.-S. Byun and L. Wang, Elliptic equations with measurable coefficients in Reifenberg domains, Adv. Math. (5) 225 (2010), 2648–2673.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. A. Caffarelli and I. Peral, On W1, p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. (1) 51 (1998), 1–21.

    Article  MATH  MathSciNet  Google Scholar 

  8. M. Giaquinta, “Introduction to Regularity Theory for Nonlinear Elliptic Systems”, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 1993.

    Google Scholar 

  9. P. Grisvard, “Singularities in Boundary Value Problems”, Vol. 22 of Recherches en Mathématiques Appliquées [Research in Applied Mathematics], Masson, Paris, 1992.

    Google Scholar 

  10. D. Jerison and C. E. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal. (1) 130, 161–219.

    Google Scholar 

  11. D. S. Jerison and C. E. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. (1) 46 (1982), 80–147.

    Article  MATH  MathSciNet  Google Scholar 

  12. C. Kkenig and T. Toro, Harmonic measure on locally flat domains, Duke Math. J. (3) 87 (1997), 509–551.

    Article  Google Scholar 

  13. C. Kenig and T. Toro, Free boundary regularity for harmonic measures and Poisson kernels, Ann. of Math. (2), (2) 150 (1999), 369–454.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Kenig and T. Toro, Poisson kernel characterization of Reifen-berg flat chord arc domains, Ann. Sci. École Norm. Sup. (4), (3) 36 (2003), 323–401.

    MATH  MathSciNet  Google Scholar 

  15. A. Lemenant, Energy improvement for energy minimizing junctions in the complement of generalized Reifenberg-flat sets Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), (2) 9 (2010), 351–384.

    MATH  MathSciNet  Google Scholar 

  16. A. Lemenant and E. Milakis, Quantitative stability for the first Dirichlet eigenvalue in Reifenberg flat domains in ℝ N, J. Math. Anal. Appl. (2) 364 (2010), 522–533.

    Article  MATH  MathSciNet  Google Scholar 

  17. A. L emenant and E. Milakis, A stability result for nonlinear Neumann problems in Reifenberg flat domains in ℝ N, Publ. Mat. (2) 55 (2011), 413–432.

    Article  MATH  MathSciNet  Google Scholar 

  18. A. Lemenant, E. Milakis and L. V. Spinolo, Spectral stability estimates for the Dirichlet and Neumann Laplacian in rough domains, J. Funct. Anal. (9) 264 (2013), 2097–2135.

    Article  MATH  MathSciNet  Google Scholar 

  19. E. Milakis and T. Toro, Divergence form operators in Reifen-berg flat domains, Math. Z. (1) 264 (2010), 15–41.

    Article  MATH  MathSciNet  Google Scholar 

  20. T. Toro, Geometry of measures: harmonic analysis meets geometric measure theory, In: “Handbook of Geometric Analysis”, No. 1, Vol. 7 of Adv. Lect. Math. (ALM), Int. Press, Somerville, MA, 2008, 449–465.

    Google Scholar 

Download references

Authors
  1. Antoine Lemenant
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yannick Sire
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. CMAP, Ecole Polytechnique, CNRS, 91128, Palaiseau Cedex, France

    Antonin Chambolle

  2. Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, 56127, Pisa, Italia

    Matteo Novaga

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Scuola Normale Superiore Pisa

About this paper

Cite this paper

Lemenant, A., Sire, Y. (2013). Boundary regularity for the Poisson equation in reifenberg-flat domains. In: Chambolle, A., Novaga, M., Valdinoci, E. (eds) Geometric Partial Differential Equations proceedings. CRM Series, vol 15. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-473-1_10

Download citation

Publish with us

Policies and ethics

Search

Navigation