We provide an approach of elastic flow with dynamic conditions at the knot points to resolve the path planning problem in the plane. The elastic flow is set up by a second-order semilinear parabolic system with nonlocal lower-order terms. We prove the existence of classical global solutions to the elastic flow, whose asymptotic limits provide classical solutions to the path planning problem.

The recent breakthrough result of Guth, Iosevich, Ou, and Wang (2019) on the Falconer distance problem states that for a compact set A ⊂ R 2 , if the Hausdorff dimension of A is greater than 5 4 , then the distance set ∆(A) has positive Lebesgue measure. Ou and Taylor (2021) recently generalized this result to the setting of (distance) trees. The main purpose of this paper is to study the discrete version of their result over both prime fields and arbitrary fields for small sets.