Serena Dipierro, Ovidiu Savin and Enrico Valdinoci
Boundary behavior of nonlocal minimal surfaces
(775K, pdf)
ABSTRACT. 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.