Strong Uniqueness for Fourth Order Elliptic Differential Operators

  • Philippe Le Borgne
Chapter
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 46)

Abstract

This article describes a new strong uniqueness result for fourth order elliptic differential operators. The main theorem extends the classical results for second order operators to fourth order differential operators which are factorized into two second order operators. Uniqueness is associated with the differential inequality
$$ \begin{gathered} \left| {P\left( {x,D} \right)u} \right| \leqslant C_1 \frac{{\left| u \right|}} {{\left| x \right|^4 }} + C_2 \frac{{\left| {\nabla u} \right|}} {{\left| x \right|^3 }} + C_3 \frac{{\left( {\sum\nolimits_{\left| \alpha \right| = 2} {\left| {D^\alpha u} \right|^2 } } \right)^{\frac{1} {2}} }} {{\left| x \right|^2 }} \hfill \\ + C_4 \frac{{\sum\nolimits_{\left| \beta \right| = 3} {\left| {D^\beta u} \right|} }} {{\left| x \right|^{1 - \varepsilon } }} \left( {\varepsilon > 0} \right), \hfill \\ \end{gathered} $$
where C 1, C 2, C 3, C 4 are positive constants and C 3 < 3/2; in addition, we suppose that P(x, D) is a differential operator with complex Lipschitz continuous coefficients and also that P(x, D) = Q 1(x, D) Q 2(x, D) where Q 1(x, D) and Q 2(x, D) are two second order differential elliptic operators such that Q 1(O, D) = Q 2(0, D) = −Δ. The proof of the theorem mentioned above uses the classical Carleman method.

Keywords

Manifold Lution Dinates Veri 

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Copyright information

© Springer Science+Business Media New York 2001

Authors and Affiliations

  • Philippe Le Borgne
    • 1
  1. 1.Département de MathématiquesUniversité de ReimsReims Cédex 2France

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