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**Unique continuation for elliptic operators: A geometric-variational approach.**
*(English)*
Zbl 0674.35007

The paper is concerned with an approach to unique continuation which is not based on the Carleman method, but rather on obtaining direct quantitative information on the order of vanishing of a solution of an elliptic pde. Such an approach was introduced by the authors [Monotonicity properties of variational integrals, \(A_ p\) weights and unique continuation, Indiana Univ. Math. J. 35, No.2, 245-268 (1986)] for the class of equations \(div(A(x)\nabla u)=0\), where \(A(x)=(a_{ij}(x))\) is a uniformly elliptic, symmetric matrix, with Lipschitz continuous entries. The results in this paper extend those of the authors (loc. cit.) to the class of equations
\[
-div(A(x)\nabla u)+\vec b(x)\circ \nabla u+V(x)u=0,
\]
where the lower order terms are allowed to be singular. This extension is based on a rather delicate analysis that ultimately relies on a strong form of uncertainty principle.

Reviewer: N.Garofalo

### MSC:

35B60 | Continuation and prolongation of solutions to PDEs |

35A15 | Variational methods applied to PDEs |

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

35J15 | Second-order elliptic equations |

### Keywords:

unique continuation; Carleman method; Monotonicity; variational integrals; uncertainty principle
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\textit{N. Garofalo} and \textit{F.-H. Lin}, Commun. Pure Appl. Math. 40, No. 3, 347--366 (1987; Zbl 0674.35007)

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### References:

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[2] | Carleman, Ark. Mat. 26B pp 1– (1939) |

[3] | Fabes, Comm. in PDE 7 pp 56– (1982) · Zbl 0498.35042 |

[4] | Garofalo, Indiana University Math. J. 35 pp 245– (1986) |

[5] | Hörmander, Comm. in PDE 8 pp 21– (1983) |

[6] | Jerison, Annals of Math. 121 pp 463– (1985) |

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[8] | and , Methods of Modern Mathematical Physics, Vols. II and IV, Academic Press, New York, 1978. |

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[10] | An appendix to Unique continuation and absence of positive eigenvalues for Schrödinger operators,” by D. Jerison and C. E. Kenig, Annals of Math., 121, 1985, pp. 489–494. |

[11] | The Theory of Groups and Quantum Mechanics, Dover, New York, 1950. |

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