Skip to main content
SpringerLink
Log in

Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients

  • Conference paper
  • First Online:
Anomalies in Partial Differential Equations

Part of the book series: Springer INdAM Series ((SINDAMS,volume 43))

Abstract

In this paper we will analyze the local solvability property of some second order linear degenerate partial differential operators with non-smooth coefficients. We will start by considering some operators with C α, 1 coefficients, with α = 0, 1, having a kind of affine structure. Next, we will study operators with a more general structure having C 0, 1 or L ∞ coefficients. In both cases the local solvability will be analyzed at multiple characteristic points where the principal symbol may possibly change sign.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover 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

Similar content being viewed by others

References

  1. Colombini, F., Cordaro, P., Pernazza, L.: Local solvability for a class of evolution equations. J. Funct. Anal. 258, 3469–3491 (2010)

    Article  MathSciNet  Google Scholar 

  2. Cicognani, M., Reissig, M.: Well-posedness for degenerate Schrödinger equations. Evol. Equ. Control Theory 3(1), 15–33 (2014)

    Article  MathSciNet  Google Scholar 

  3. Dencker, N.: Solvability and limit bicharacteristics. J. Pseudo-Differ. Oper. Appl. 7(3), 295–320 (2016). https://doi.org/10.1007/s11868-016-0164-x

    Article  MathSciNet  Google Scholar 

  4. Federico, S.: A model of solvable second order PDE with non smooth coefficients. J. Math. Anal. Appl. 440(2), 661–676 (2016). https://doi.org/10.1016/j.jmaa.2016.03.056

    Article  MathSciNet  Google Scholar 

  5. Federico, S.: Local Solvability of a Class of Degenerate Second Order Operators. Ph.D. thesis, University of Bologna (2017)

    Google Scholar 

  6. Federico, S.: Sufficient conditions for local solvability of some degenerate PDO with complex subprincipal symbol. J. Pseudo-Differ. Oper. Appl. 10(4), 929–940 (2019). https://doi.org/10.1007/s11868-018-0264-x

    Article  MathSciNet  Google Scholar 

  7. Federico, S., Parmeggiani, A.: Local solvability of a class of degenerate second order operators. Commun. Partial Differ. Equ. 41(3), 484–514 (2016). https://doi.org/10.1080/03605302.2015.1123273

    Article  MathSciNet  Google Scholar 

  8. Federico, S., Parmeggiani, A.: Local solvability of a class of degenerate second order operators. Commun. Partial Differ. Equ. 43(10), 1485–1501 (2018). https://doi.org/10.1080/03605302.2018.1517789

    Article  MathSciNet  Google Scholar 

  9. Federico, S., Staffilani, G.: Smoothing effect for time-degenerate Schrödinger operators. Preprint. Arxiv

    Google Scholar 

  10. Helffer, B.: Construction de paramétrixes pour des opérateurs pseudodifférentiels caractéristiques sur la réunion de deux cônes lisses. Bull. Soc. Math. France Suppl. Mém. 51–52, 63–123 (1977)

    Article  Google Scholar 

  11. Hörmander, L.: The Analysis of Linear Partial Differential Operators. III. Pseudodifferential Operators. Grundlehren der Mathematischen Wissenschaften, vol. 274, viii+ 525 pp. Springer, Berlin (1985)

    Google Scholar 

  12. Hörmander, L.: The analysis of linear partial differential operators. IV. Fourier integral operators. Grundlehren der Mathematischen Wissenschaften, vol. 275, vii+ 352 pp. Springer, Berlin (1985)

    Google Scholar 

  13. Kannai, Y.: An unsolvable hypoelliptic differential operator. Israel J. Math. 9, 306–315 (1971)

    Article  MathSciNet  Google Scholar 

  14. Kohn, J.J.: Loss of derivatives. In: From Fourier Analysis and Number Theory to Radon Transforms and Geometry. Dev. Math., vol. 28, pp. 353–369. Springer, New York (2013)

    Google Scholar 

  15. Lerner, N.: When is a pseudo-differential equation solvable? Ann. Inst. Fourier 50, 443–460 (2000)

    Article  MathSciNet  Google Scholar 

  16. Lerner, N.: Metrics on the Phase Space and Non-Selfadjoint Pseudo-Differential Operators. Pseudo-Differential Operators. Theory and Applications, vol. 3, xii+ 397 pp. Birkhäuser, Basel (2010)

    Google Scholar 

  17. Mendoza, G.A.: A necessary condition for solvability for a class of operators with involutive double characteristics. In: Microlocal Analysis (Boulder, Colo., 1983), vol. 27, pp. 193–197. Contemporary Mathematics. American Mathematical Society, Providence (1984)

    Google Scholar 

  18. Mendoza, G.A., Uhlmann, G.A.: A necessary condition for local solvability for a class of operators with double characteristics. J. Funct. Anal. 52, 252–256 (1983)

    Article  MathSciNet  Google Scholar 

  19. Müller, D.: Local solvability of linear differential operators with double characteristics. II. Sufficient conditions for left-invariant differential operators of order two on the Heisenberg group. J. Reine Angew. Math. 607, 1–46 (2007)

    MATH  Google Scholar 

  20. Müller, D.: Local solvability of linear differential operators with double characteristics. I. Necessary conditions. Math. Ann. 340, 23–75 (2008)

    MathSciNet  Google Scholar 

  21. Müller, D., Ricci, F.: Solvability for a class of doubly characteristic differential operators on 2-step nilpotent groups. Ann. Math. 143, 1–49 (1996)

    Article  MathSciNet  Google Scholar 

  22. Müller, D., Ricci, F.: Solvability for a class of non-homogeneous differential operators on two-step nilpotent groups. Math. Ann. 304, 517–547 (1996)

    Article  MathSciNet  Google Scholar 

  23. Parenti, C., Parmeggiani, A.: On the solvability of a class of ψdos with multiple characteristics. Int. Math. Res. Not. 2014(14), 3790–3817 (2014). https://doi.org/10.1093/imrn/rnt061

    Article  MathSciNet  Google Scholar 

  24. Parmeggiani, A.: On the solvability of certain degenerate partial differential operators. In: Colombini, F., Del Santo, D., Lannes, D. (eds.), Shocks, Singularities and Oscillations in Nonlinear Optics and Fluid Mechanics. Springer INdAM Series, vol. 17, pp. 151–179. Springer, Berlin (2017). https://doi.org/10.1007/978-3-319-52042-1

    Google Scholar 

  25. Peloso, M., Ricci, F.: Analysis of the Kohn Laplacian on quadratic CR manifolds. J. Funct. Anal. 203, 321–355 (2003)

    Article  MathSciNet  Google Scholar 

  26. Treves, F.: On a question of Louis Nirenberg. Math. Res. Lett. 10, 729–735 (2003)

    Article  MathSciNet  Google Scholar 

  27. Treves, F.: On the solvability and hypoellipticity of complex vector fields. In: Geometric Analysis of Several Complex Variables and Related Topics. Contemporary Mathematics, vol. 550, pp. 173–196. American Mathematical Society, Providence, (2011)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Ghent, Belgium

    Serena Federico

Authors
  1. Serena Federico
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Serena Federico .

Editor information

Editors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Massimo Cicognani

  2. Dipartimento di Matematica e Geoscienze, Università di Trieste, Trieste, Italy

    Daniele Del Santo

  3. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Alberto Parmeggiani

  4. Institut für Angewandte Analysis, TU Bergakademie Freiberg, Freiberg, Germany

    Michael Reissig

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Federico, S. (2021). Local Solvability of Some Partial Differential Operators with Non-smooth Coefficients. In: Cicognani, M., Del Santo, D., Parmeggiani, A., Reissig, M. (eds) Anomalies in Partial Differential Equations. Springer INdAM Series, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-030-61346-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation