We prove that nonlocal minimal graphs in the plane exhibit generically stickiness effects and boundary discontinuities. More precisely, we show that if a nonlocal minimal graph in a slab is continuous up to the boundary, then arbitrarily small perturbations of the far-away data produce boundary discontinuities. Hence, either a nonlocal minimal graph is discontinuous at the boundary, or a small perturbation of the prescribed conditions produces boundary discontinuities. The proof relies on a sliding method combined with a fine boundary regularity analysis, based on a discontinuity/smoothness alternative. Namely, we establish that nonlocal minimal graphs are either discontinuous at the boundary or their derivative is Hölder continuous up to the boundary. In this spirit, we prove that the boundary regularity of nonlocal minimal graphs in the plane “jumps” from discontinuous to $$C^{1,\gamma }$$ C 1 , γ , with no intermediate possibilities allowed. In particular, we deduce that the nonlocal curvature equation is always satisfied up to the boundary. As an interesting byproduct of our analysis, one obtains a detailed understanding of the “switch” between the regime of continuous (and hence differentiable) nonlocal minimal graphs to that of discontinuous (and hence with differentiable inverse) ones.

中文翻译：

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平面中的非局部极小图是一般粘性的

我们证明平面中的非局部极小图表现出一般的粘性效应和边界不连续性。更准确地说，我们表明，如果平板中的非局部极小图在边界上是连续的，那么远处数据的任意小扰动会产生边界不连续性。因此，非局部极小图在边界处是不连续的，或者规定条件的小扰动会产生边界不连续性。证明依赖于滑动方法结合精细边界规律分析，基于不连续性/平滑度替代方案。也就是说，我们确定非局部极小图要么在边界处不连续，要么它们的导数在边界上是 Hölder 连续的。本着这种精神，我们证明平面中非局部极小图的边界正则性从不连续“跳跃”到 $$C^{1,\gamma}$$C 1 , γ ，不允许中间可能性。特别地，我们推导出非局部曲率方程总是满足直到边界。作为我们分析的一个有趣的副产品，人们详细了解了连续（因此可微）非局部极小图的状态与不连续（因此具有可微逆）的状态之间的“转换”。