Aug 27, 2020
Webinar – Mathematical Research Seminar Series
Zibu Liu, Duke University
Title: Dislocation, high-dimensional Peierls-Nabarro model and the De-Giorgi conjecture: existence and rigidity
Abstract: In this talk, I will introduce the existence and rigidity of the vectorial Peierls-Nabarro (PN) model for dislocations in high dimensions. From the original vectorial PN model, we first derive a non-local scalar Ginzburg-Landau equation with an anisotropic positive (if Poisson ratio belongs to $(-1/2,1/3)$) singular kernel. Utilizing energy decreasing rearrangement method, existence of minimizers of the PN energy is verified. For rigidity, a De Giorgi-type conjecture of single-variable symmetry for solutions is established using a method of spectral analysis. This method is powerful for nonlocal pseudo-differential operators with strong maximal principle. The physical interpretation of this rigidity result is that provided exclusive dependence of the misfit potential on the shear displacement, the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile. This is a joint work with Jian-Guo Liu and Yuan Gao, both from Duke University.
Slides for the talk (click)
Recorded video for the talk (click)
