2012/12/17 by Blair Davey, Davey, Blair
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems #Differential Equations and Boundary Problems
paper · pdf · doi:10.48550/arxiv.1212.4085
We prove sharpness of quantitative unique continuation results for solutions of -Δu + W⋅ ∇ u + V u = \la u, where \la ∈ \C and V and W are complex-valued decaying potentials that satisfy |V(x)| \lesssim -N and |W(x)| \lesssim -P. For M(R) = inf|x0| = R||u||L2(B1(x0)), it was shown in a companion paper that if the solution u is non-zero, bounded, and u(0) = 1, then M(R) \gtrsim exp(-C R\be0(log R)A(R)), where \be0 = max2 - 2P, (4-2N)/3, 1. Under certain conditions on N, P, \la, and the dimension, we construct examples (some of which are in the style of Meshkov) to prove that this estimate for M(R) is sharp. That is, we construct functions u, V and W such that -Δu + W⋅ ∇ u + V u = \la u, |V(x)| \lesssim -N, |W(x)| \lesssim -P and |u(x)| \lesssim exp(-c|x|\be0(log |x|)C).