Quantitative uniqueness of solutions to second-order elliptic equations with singular lower order terms
Blair Davey, Jiuyi Zhu · Communications in Partial Differential Equations · 2019
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 Δu+W·∇u+Vu=0 in some open, connected subset of Rn, where n≥3, 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 Δu+W·∇u+Vu=0 in Rn. The main tools in our work are new versions of Lp→Lq Carleman estimates for a range of p and q values.