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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 $div\left(A\left(x\right)\nabla u\right)=0$, where $A\left(x\right)=\left({a}_{ij}\left(x\right)\right)$ 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

$-div\left(A\left(x\right)\nabla u\right)+\stackrel{\to }{b}\left(x\right)\circ \nabla u+V\left(x\right)u=0,$

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 of solutions of PDE 35A15 Variational methods (PDE) 35A30 Geometric theory for PDE, characteristics, transformations 35J15 Second order elliptic equations, general