2019/03/10 by Davey, Blair · 1 citation
#35A02 #35J10 #35J15 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1903.04021
We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for Δ+ V. That is, for any non-trivial u that solves Δu + V u = 0 in some open, connected subset of ℝn, we estimate the vanishing order of solutions in terms of the Lt-norm of V. Our results apply to all t > \frac n 2 and n ≥ 3. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to Δu + V u = 0. To handle V ∈ Lt for every t ∈ (\frac n 2, ∞], we prove a novel Lp - Lq Carleman estimate by interpolating a known Lp - L2 estimate with a new endpoint Carleman estimate. This new Carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term, those of the form Δu + W ⋅ ∇ u + V u = 0.