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

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 $$\Delta u= V u$$.