Quantitative uniqueness of solutions to second-order elliptic equations with singular lower order terms

Abstract

In this article, we study some quantitative unique continuation properties of solutions to second-order elliptic equations with singular lower order terms. First, we quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when u is a nontrivial solution to in some open, connected subset of where we characterize the vanishing order of solutions in terms of the norms of V and W in their respective Lebesgue spaces. Then, using these maximal order of vanishing estimates, we establish quantitative unique continuation at infinity results for solutions to in The main tools in our work are new versions of Carleman estimates for a range of p and q values.