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Existence and rigidity of the vectorial Peierls-Nabarro model for dislocations in high dimensions. (English) Zbl 1478.35199

Authors’ abstract: We focus on the existence and rigidity problems of the vectorial Peierls-Nabarro (PN) model for dislocations. Under the assumption that the misfit potential on the slip plane only depends on the shear displacement along the Burgers vector, a reduced non-local scalar Ginzburg-Landau equation with an anisotropic positive (if Poisson ratio belongs to \((-1/2, 1/3)\)) singular kernel is derived on the slip plane. We first prove that minimizers of the PN energy for this reduced scalar problem exist. Starting from \(H^{1/2}\) regularity, we prove that these minimizers are smooth 1D profiles only depending on the shear direction, monotonically and uniformly converge to two stable states at far fields in the direction of the Burgers vector. Then a De Giorgi-type conjecture of singlevariable symmetry for both minimizers and layer solutions is established. As a direct corollary, minimizers and layer solutions are unique up to translations. The proof of this De Giorgi-type conjecture relies on a delicate spectral analysis which is especially powerful for nonlocal pseudo-differential operators with strong maximal principle. All these results hold in any dimension since we work on the domain periodic in the transverse directions of the slip plane. The physical interpretation of this rigidity result is that the equilibrium dislocation on the slip plane only admits shear displacements and is a strictly monotonic 1D profile provided exclusive dependence of the misfit potential on the shear displacement.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35Q56 Ginzburg-Landau equations
74B10 Linear elasticity with initial stresses
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35S15 Boundary value problems for PDEs with pseudodifferential operators
35J50 Variational methods for elliptic systems
35B65 Smoothness and regularity of solutions to PDEs
35B50 Maximum principles in context of PDEs
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