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

2017/02/15 by Blair Davey, Davey, Blair, Jiuyi Zhu +1 · 3 citations
Computer Science · Mathematics · #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1702.04742

openalex publication_date 2017/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article, we study the quantitative uniqueness of solutions to second order elliptic equations with singular lower order terms. We quantify the strong unique continuation property by estimating the maximal vanishing order of solutions. That is, when u is a non-trivial solution to \triangle u + W ⋅ ∇ u + V u = 0 in some open, connected subset of \mathbb 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. Using these maximal order of vanishing estimates, we also establish quantitative unique continuation at infinity results for solutions to \triangle u + W ⋅ ∇ u + V u = 0 in \mathbb Rⁿ. The main tools in our work are new versions of L^p→ L^q Carleman estimates for a range of p- and q-values.

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