# 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

Differential Operator Differential Inequality Order Operator Unique Continuation Strong Uniqueness
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.

## References

1. 
S. Alinhac, Non-unicité pour des opérateurs différentiels à caractéristiques complexes simples, Ann. Sci. Ec. Norm. Sup. 13 (1980), 385–393.
2. 
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.
3. 
S. Alinhac and N. Lerner, Unicité forte partir d’une variété de dimension quelconque pour des inégalités différentielles elliptiques, Duke Math. Journal 48 (1981), 49–68.
4. 
N. Aronszajn, A unique continuation theorem for solutions of elliptic partial differential equations or inequalities of second order, J. Math. Pures Appl. 36 (1957), 235–249.
5. 
N. Aronszajn, A. Krzywicki, and J. Szarski, A unique continuation theorem for exterior differential forms on Riemannian manifolds, Ark. for Mat. 4 (1962), 417–453.
6. 
T. Carleman, Sur un problème d’unicité pour les systèmes d’équations aux dérivées partielles a deux variables indépendantes, Ark. for Mat. 26 B. 17 (1938), 1–9.Google Scholar
7. 
F. Colombini and C. Grammatico, A counterexample to strong uniqueness for all powers of the Laplace operator, Comm. Partial Differential equations 25(2000), 585–600.
8. 
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.
9. 
H.O. Cordes, Uber die bestimmheit des losungen elliptischer differentialgleichungen durch anfangsvorgaben, Nachr. Akad. Wiss. Gottingen Math. Phys. Kl. IIa 11 (1956), 239–258.
10. 
P.M. Goorjian, The uniqueness of the Cauchy problem for partial differential equations which may have multiple characteritics, Trans. Amer. Math. Soc. 146 (1969), 493–509.
11. 
C. Grammatico, A result on strong unique continuation for the Laplace operator, Comm. Partial Differential Equations 22 (1997), 1475–1491.
12. 
L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, 1963.Google Scholar
13. 
L. Hörmander, On the uniqueness of the Cauchy problem I-II. Math. Scand. 6 (1958), 213–225; 7 (1959), 177–190.
14. 
L. Hörmander, The Analysis of Linear Partial Differential Operators, 3, Springer-Verlag, 1985.
15. 
L. Hörmander, The Analysis of Linear Partial Differential Operators, 4, Springer-Verlag, 1985.
16. 
L. Hörmander, Uniqueness theorems for second order elliptic differential equations, Comm. Partial Differential Equations 8 (1983), 21–64.
17. 17.
[] P. Le Borgne, Unicité forte pour le produit de deux opérateurs elliptiques d’ordre 2, prépublication, Département de Mathématiques, Université de Reims, (1998), to appear in Indiana Univ. Math. J. Google Scholar
18. 
N. Lerner, Résultats d’unicité forte pour des opérateurs elliptiques h coefficients Gevrey, Comm. Partial Differential Equations 6 (1981), 1163–1177.
19. 
R.N. Pederson, On the unique continuation theorem for certain second and fourth order elliptic equations, Comm. Pure Appl. Math. 11 (1958), 67–80.
20. 
R.N. Pederson, Uniqueness in Cauchy’s problem for equations with double characteristics. Ark. Math. 6 (1967), 535–549.
21. 
A. Plia, A smooth linear elliptic differential equation without any solution in a sphere, Comm. Pure and Appl. Math. 14 (1961), 599–617.Google Scholar
22. 
M.H. Protter, Unique continuation for elliptic equations, Trans. Amer. Math. Soc. 95 (1961), 81–91.
23. 
R. Regbaoui, Strong unique continuation for second order elliptic differential operators, Journal of Differential Equations 141(2) (1997), 201–217.
24. 
T. Shirota, A remark on the unique continuation theorem for certain fourth order elliptic equations, Proc. Jap. Acad. 36 (1960), 571–573.
25. 
C. Zuily, Uniqueness and non uniqueness in the Cauchy Problem, Progress in Mathematics 33, Birkäuser, 1983.Google Scholar 