11 research outputs found
Fractal solutions of linear and nonlinear dispersive partial differential equations
In this paper we study fractal solutions of linear and nonlinear dispersive
PDE on the torus. In the first part we answer some open questions on the
fractal solutions of linear Schr\"odinger equation and equations with higher
order dispersion. We also discuss applications to their nonlinear counterparts
like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation
(KdV).
In the second part, we study fractal solutions of the vortex filament
equation and the associated Schr\"odinger map equation (SM). In particular, we
construct global strong solutions of the SM in for for which
the evolution of the curvature is given by a periodic nonlinear Schr\"odinger
evolution. We also construct unique weak solutions in the energy level. Our
analysis follows the frame construction of Chang {\em et al.} \cite{csu} and
Nahmod {\em et al.} \cite{nsvz}.Comment: 28 page
REMOVABLE SETS FOR LIPSCHITZ HARMONIC FUNCTIONS ON CARNOT GROUPS
Abstract. Let G be a Carnot group with homogeneous dimension Q ≥ 3 and let L be a sub-Laplacian on G. We prove that the critical dimension for removable sets of Lipschitz L-harmonic functions is (Q − 1). Moreover we construct self-similar sets with positive and finite H Q−1 measure which are removable. 1