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Strong Uniqueness for Laplace and Bi-Laplace Operators in the Limit Case

  • F. Colombini
  • C. Grammatico
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)

Abstract

In this article we study some limiting cases of strong unique continuation for inequalities of the type
$$ \left| {\Delta u\left( x \right)} \right| \leqslant \frac{A} {{\left| x \right|^2 }}\left| {u\left( x \right)} \right| + \frac{B} {{\left| x \right|}}\left| {\nabla u\left( x \right)} \right| x \in \Omega , $$
(1.1)
or
$$ \left| {\Delta ^2 u\left( x \right)} \right| \leqslant \frac{A} {{\left| x \right|^4 }}\left| {u\left( x \right)} \right| + \frac{B} {{\left| x \right|^3 }}\left| {\nabla u\left( x \right)} \right| + \frac{C} {{\left| x \right|^2 }}\sum\limits_{i,j} {\left| {D_i D_j u\left( x \right)} \right|} x \in \Omega , $$
(1.2)
where Ω is a neighbourhood of the origin in R n , and A, B, C are positive constants.

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • F. Colombini
    • 1
  • C. Grammatico
    • 2
  1. 1.Dipartimento di MatematicaUniversità di PisaPisaItaly
  2. 2.Dipartimento di MatematicaUniversitä di BolognaBolognaItaly

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