Strong Unique Continuation Property for First Order Elliptic Systems

  • Takashi Ōkaji
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)


There is a long history about the strong unique continuation property, going back to the works of T. Carleman, C. Müller, E. Heinz, N. Aronszajn and H. O. Cordes. After their works many advances were made, among them, differential inequalities with critical singularities as well as subcritical ones were intensively investigated in connection with the absence of positive eigenvalues in the continuous spectrum([2], [15], [16], [17], [8], [4]).


Dirac Equation Dirac Operator Elliptic System Differential Inequality Infinite Order 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Alinhac and M.S. Baouendi, Uniqueness for the characteristic Cauchy problem and strong unique continuation for higher order partial differential inequalities Amer. J. Math.102 (1980), 179–217.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    S. Alinhac and M.S. Baouendi, A counterexample to strong uniqueness for partial differential equations of Schrödinger type Partial Diff. Eq., 19 (1994), 1727–1733.MathSciNetzbMATHCrossRefGoogle Scholar
  3. T. Carleman, Sur un problém d’unicité pour les systemes d’équations aux dérivées partielles h deux variables indépendantes Arkiv for Matematik Astr. Fys.26B (1939), 1–9.MathSciNetGoogle Scholar
  4. [4]
    F. Colombini and C. Grammatico, Some remarks on strong unique continuation for the Laplace operator and its powers Comm.Partial Differential Equations, 24(5–6) (1999), 1079–1094.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    A, Douglis, Uniqueness in Cauchy problems for elliptic systems of equations Comm.Pure Appl. Math., 6 (1953), 291–298.Google Scholar
  6. [6]
    L. De Carli and T. Okaji, Strong unique continuation property for the Dirac equation Publ. RIMS Kyoto Univ.35(6) (1999), 825–846.zbMATHCrossRefGoogle Scholar
  7. [7]
    G.N. Hile and M.H. Protter, Unique continuation and the Cauchy problem for first order systems of partial differential equations Comm. P.D.E. 1 (1976), 437–465.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    C. Grammatico, Unicitá forte per operatori ellittici, Tesi di Dottorato, Univ. degli Studi di Pisa, 1997.Google Scholar
  9. [9]
    D. Jerison, Carleman inequalities for the Dirac and Laplace operator and unique continuation Adv. Math.63 (1986), 118–134.MathSciNetCrossRefGoogle Scholar
  10. [10]
    H. Kalf and O. Yamada, Note on the paper by De Carli and Colcaji on the strong unique continuation property for the Dirac equation Publ. RIMS Kyoto Univ., 35(6) (1999), 847–852.Google Scholar
  11. [11]
    T. Okaji, Uniqueness of the Cauchy problem for elliptic operators with fourfold characteristics of constant multiplicity Comm. P.D.E. 22(1 2) (1997), 269–305.MathSciNetGoogle Scholar
  12. [12]
    T. Okaji, Strong unique continuation property for time harmonic Maxwell equations, to appear in J. Math. Soc. Japan. Google Scholar
  13. [13]
    T. Olcaji, Strong unique continuation property for elliptic systems of normal type in two independent variables, preprint.Google Scholar
  14. [14]
    Y. F. Pan, Unique continuation for Schrödinger operators with singular potentials Comm.Partial Differential Equations17 (1992), 953–965.zbMATHCrossRefGoogle Scholar
  15. [15]
    R. Regbaoui, Strong unique continuation results for differential inequalities J. Punct. Anal.148 (1997), 508–523.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [16]
    R. Regbaoui, Strong unique continuation for second order elliptic differential operators J. Diff. Eqs.141 (1997), 201–217.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [17]
    R. Regbaoui, Strong unique continuation for second order elliptic differential operators, Séminaire sur les équations aux dérivées partielles, 1996–1997, Exp. No. III, Ecole Polytech., Palaiseau, 1997.Google Scholar
  18. [18]
    V. Vogelsang, Absence of embedded eigenvalues of the Dirac equation for long range potentials Analysis 7 (1987), 259–274.MathSciNetzbMATHGoogle Scholar
  19. [19]
    V. Vogelsang, On the strong continuation principle for inequalities of Maxwell type Math. Ann.289 (1991) 285–295.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Takashi Ōkaji
    • 1
  1. 1.Division of Mathematics Graduate School of ScienceKyoto UniversityKyotoJapan

Personalised recommendations