论文标题
PEIERLS-NABARRO模型的存在和刚性，用于高维的位错
Existence and rigidity of the Peierls-Nabarro model for dislocations in high dimensions
论文作者
论文摘要
我们专注于媒介物PEIERLS-NABARRO（PN）模型的存在和僵化问题。在假设滑移平面上的失配电位仅取决于沿汉堡矢量的剪切位移的假设，具有各向异性阳性的非本地标量银堡 - landau方程降低（如果Poisson比率属于$（-1/2,1/2,1/3）$（如果属于$（-1/2,1/3）$）$）$（-1/2,1/3）$）$ sickular sickular nikular nikular on the Slip Plane。我们首先证明存在此减少标量问题的PN能量的最小化器。从$ h^{1/2} $规律性开始，我们证明这些最小化器仅取决于剪切方向，这是单调和均匀地收敛到伯格斯矢量方向的远场的两个稳定状态。然后，建立了对最小化器和层溶液的单变量对称性的De Giorgi构想。作为直接推论，最小化和层解决方案是独特的翻译。这种de giorgi型猜想的证明依赖于精致的光谱分析，该分析对于具有强大最大原理的非本地伪差异算子特别有力。所有这些结果都在任何维度上都存在，因为我们在滑动平面的横向方向上进行了域周期性。这种刚性结果的物理解释是，滑动平面上的平衡位错仅接受剪切位移，并且是严格的单调1D轮廓，提供了对剪切位移的独特依赖性。
We focus on 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.
