Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions
Existence and rigidity of the vectorial Peierls–Nabarro model for dislocations in high dimensions
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DOI：
10.1088/1361-6544/ac24e3
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发表时间：
2020-06
期刊：
影响因子：
1.7
通讯作者：
Yuan Gao;Jian‐Guo Liu;Zibu Liu
中科院分区：
文献类型：
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作者：
Yuan Gao;Jian‐Guo Liu;Zibu Liu
我们重点关注位错矢量 Peierls-Nabarro (PN) 模型的存在性和刚性问题。假设滑移面上的失配势仅取决于沿 Burgers 矢量的剪切位移，在滑移面上推导了具有各向异性正值（如果泊松比属于（−1/2, 1/3））奇异核的简化非局部标量 Ginzburg-Landau 方程。我们首先证明这个降标量问题的 PN 能量最小化存在。从 H 1/2 正则性出发，我们证明这些极小化器是仅取决于剪切方向的平滑一维轮廓，在 Burgers 矢量方向的远场处单调且均匀地收敛到两个稳定状态。然后建立了极小值和层解的单变量对称性的 De Giorgi 型猜想。作为直接推论，最小化器和分层解决方案在翻译方面是独一无二的。这种 De Giorgi 型猜想的证明依赖于精细的谱分析，这对于具有强极大值原理的非局部伪微分算子尤其有效。所有这些结果在任何维度上都成立，因为我们研究的是滑移面横向上的周期域。该刚性结果的物理解释是，滑移面上的平衡位错仅允许剪切位移，并且是严格单调的一维轮廓，提供了错配势对剪切位移的唯一依赖性。
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 single-variable 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.