Boundary behavior of nonlocal minimal surfaces
Boundary behavior of nonlocal minimal surfaces
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DOI：
10.1016/j.jfa.2016.11.016
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发表时间：
2015-06
影响因子：
1.7
通讯作者：
S. Dipierro;O. Savin;E. Valdinoci
中科院分区：
文献类型：
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作者：
S. Dipierro;O. Savin;E. Valdinoci
我们考虑非局部极小曲面在边界附近的行为。通过一系列详细的例子，我们表明，即使当区域是光滑和凸的时，非局部极小曲面也可能粘在区域的边界上。这是一种纯粹的非局部现象，与经典极小曲面的边界性质形成鲜明对比，特别是当球外基准面为小半环时，对半球表现出粘性现象，当左右基准面之间的摆动足够大时，对二维盒的边部表现出粘性现象，当分数维参数较小时，粘性效应会越来越明显。此外，我们还证明了平面上的线在边界处是不稳定的：即，线的小的紧支撑扰动导致板中的极小值以与扰动的幂成比例的量粘在边界上。在所有的例子中，我们都对粘性现象给出了具体的估计。此外，我们还构造了一族具有独立利益的紧支障碍。
We consider the behavior of the nonlocal minimal surfaces in the vicinity of the boundary. By a series of detailed examples, we show that nonlocal minimal surfaces may stick at the boundary of the domain, even when the domain is smooth and convex. This is a purely nonlocal phenomenon, and it is in sharp contrast with the boundary properties of the classical minimal surfaces.In particular, we show stickiness phenomena to half-balls when the datum outside the ball is a small half-ring and to the side of a two-dimensional box when the oscillation between the datum on the right and on the left is large enough.When the fractional parameter is small, the sticking effects may become more and more evident. Moreover, we show that lines in the plane are unstable at the boundary: namely, small compactly supported perturbations of lines cause the minimizers in a slab to stick at the boundary, by a quantity that is proportional to a power of the perturbation.In all the examples, we present concrete estimates on the stickiness phenomena. Also, we construct a family of compactly supported barriers which can have independent interest.