Abstract
In this chapter, we introduce nonlocal minimal surfaces. We first discuss a Bernstein type result in any dimension, namely the property that an s-minimal graph in \(\mathbb{R}^{n+1}\) is flat (if no singular cones exist in dimension n); we will then prove that an s-minimal surface whose prescribed data is a subgraph, is itself a subgraph. The non-existence of nontrivial s-minimal cones in dimension 2 is then proved. Moreover, some boundary regularity properties will be discussed at the end of this chapter: quite surprisingly, and differently from the classical case, nonlocal minimal surfaces do not always attain boundary data in a continuous way (not even in low dimension). A possible boundary behavior is, on the contrary, a combination of stickiness to the boundary and smooth separation from the adjacent portions.
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