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4 - On localization and quantitative uniqueness for elliptic partial differential equations

Published online by Cambridge University Press:  15 August 2019

Charles L. Fefferman
Affiliation:
Princeton University, New Jersey
James C. Robinson
Affiliation:
University of Warwick
José L. Rodrigo
Affiliation:
University of Warwick
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Summary

We address the decay and the quantitative uniqueness properties for solutions of the elliptic equation with a gradient term, $$\Delta u=W\cdot \nabla u$$. We prove that there exists a solution in a complement of the unit ball which satisfies $$|u(x)|\le C\exp (-C^{-1}|x|^2)$$ where $$W$$ is a certain function bounded by a constant. Next, we revisit the quantitative uniquenessfor the equation$$-\Delta u= W \cdot \nabla u$$ and provide an example of a solution vanishing at a point with the rate$${\rm const}\Vert W\Vert_{L^\infty}^2$$.We also review decay and vanishing results for the equation $$\Delta u= V u$$.

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Publisher: Cambridge University Press
Print publication year: 2018

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