Existence and uniqueness of bounded stable solutions to the Peierls-Nabarro model for curved dislocations. (English) Zbl 1462.35437
Summary: We study the well-posedness of the vector-field Peierls-Nabarro model for curved dislocations with a double well potential and a bi-states limit at far field. Using the Dirichlet to Neumann map, the 3D Peierls-Nabarro model is reduced to a nonlocal scalar Ginzburg-Landau equation. We derive an integral formulation of the nonlocal operator, whose kernel is anisotropic and positive when Poisson’s ratio \(\nu \in (-\frac{1}{2}, \frac{1}{3})\). We then prove that any bounded stable solution to this nonlocal scalar Ginzburg-Landau equation has a 1D profile, which corresponds to the PDE version of flatness result for minimal surfaces with anisotropic nonlocal perimeter. Based on this, we finally obtain that steady states to the nonlocal scalar equation, as well as the original Peierls-Nabarro model, can be characterized as a one-parameter family of straight dislocation solutions to a rescaled 1D Ginzburg-Landau equation with the half Laplacian.
MSC:
| 35R11 | Fractional partial differential equations |
| 35Q56 | Ginzburg-Landau equations |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35J50 | Variational methods for elliptic systems |
| 35J60 | Nonlinear elliptic equations |
Keywords:
Dirichlet to Neumann map; nonlocal scalar Ginzburg-Landau equation; latness of minimal surfacesReferences:
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