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Publications [#355948] of Yuan Gao
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 Dong, H; Gao, Y, Existence and uniqueness of bounded stable solutions to the Peierls–Nabarro model for curved dislocations,
Calculus of Variations and Partial Differential Equations, vol. 60 no. 2
(April, 2021) [doi]
(last updated on 2022/08/06)
Abstract: We study the wellposedness of the vectorfield Peierls–Nabarro model for curved dislocations with a double well potential and a bistates 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 ν∈(12,13). 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 oneparameter family of straight dislocation solutions to a rescaled 1D Ginzburg–Landau equation with the half Laplacian.


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