Unique Continuation from Sets of Positive Measure

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


Let Ω be a connected open subset of ℝ n and let V, W be functions on Ω. We say that the differential inequality
$$\left| {\Delta u} \right| \leqslant \left| {Vu} \right| + \left| {W\nabla u} \right|$$
has the weak unique continuation property (w.u.c.p) if any solution u of (1.1) which vanishes on an open subset of Ω is identically zero. And we say that (1.1) has the strong unique continuation property (s.u.c.p) if any solution u is identically zero whenever it vanishes of infinite order at a point of Ω. We recall that a function \(u \in L_{{loc}}^{p}\) is said to vanish of infinite order at a point x 0 (or that u is flat at x 0) if for all N > 0,
$$\int_{{\left| {x - {{x}_{0}}} \right|}} {{{{\left| {u\left( x \right)} \right|}}^{p}}dx = O\left( {{{R}^{N}}} \right)asR \to 0} .$$


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    B. Bercelo, C. Kenig, A. Ruiz and C.D. Sogge, Weighted Sobolev inequalities and unique continuation for the Laplacian plus lower order terms. III. J. Math. 32 (1988), 230–245.Google Scholar
  2. [2]
    D. De Figueiredo and J-P. Gossez, Strict monotonicity of eigenvalues and unique continuation, Comm. Partial Differential Equations 17 (1992), 339–346.zbMATHCrossRefGoogle Scholar
  3. [3]
    J-P. Gossez and A. Loulit, A note on two notions of unique continuation. Bull. Soc. Math. Belg. Ser. B 45(3) (1993), 257–268.MathSciNetzbMATHGoogle Scholar
  4. [4]
    L. Hörmander, Uniqueness theorems for second order elliptic differential equations. Comm. Partial Differential Equations 8 (1983), 21–64.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    D. Jerison and C. Kenig, Unique continuation and absence of positive eigenvalues for Schrödinger operators. Ann. of Math. 121 (1985), 463–494.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [6]
    C. Kenig, A. Ruiz and C. Sogge, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators. Duke Math. J. bf 55 (1987), 329–347.MathSciNetzbMATHGoogle Scholar
  7. [7]
    O. Ladyzenskaya and N. Uraltzeva, Linear and Quasilinear Elliptic Equations. Academic Press, New York, 1968.Google Scholar
  8. [8]
    R. Regbaoui, Strong unique continuation results for differential inequalities. J. Funct. Anal.148 (1997), 508–523.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    R. Regbaoui, Unique continuation for differential equations of Schrödinger’s type. Comm. Anal. Geom. 7 (1999), 303–323.MathSciNetzbMATHGoogle Scholar
  10. [10]
    T. Wolff, Unique continuation for and related problems. Revista Math. Iberoamericana. 6 (1990), 155–200.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    T. Wolff, A property of measures in R n and an application to unique continuation. Geom. Funct. Anal. 2 (1992), 225–284.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    T. Wolff, A counterexample in a unique continuation problem. Comm. Anal. Geom. 2 (1994), 79–102.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Rachid Regbaoui
    • 1
  1. 1.Département de MathématiquesUniversité de BrestBrestFrance

Personalised recommendations