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Zbl 0674.35007
Garofalo, Nicola; Lin, Fang-Hua
Unique continuation for elliptic operators: A geometric-variational approach. (English)
[J] Commun. Pure Appl. Math. 40, No.3, 347-366 (1987). ISSN 0010-3640

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, $A\sb p$ 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 $A(x)=(a\sb{ij}(x))$ 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.
[N.Garofalo]

MSC 2000:: *35B60 Continuation of solutions of PDE
35A15 Variational methods (PDE)
35A30 Geometric theory for PDE, transformations
35J15 Second order elliptic equations, general

Keywords: unique continuation; Carleman method; Monotonicity; variational integrals; uncertainty principle

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