We consider the singular set in the thin obstacle problem with weight
for
,
which arises as the local extension of the obstacle problem for the fractional
Laplacian (a nonlocal problem). We develop a refined expansion of the solution
around its singular points by building on the ideas introduced by Figalli and
Serra to study the fine properties of the singular set in the classical obstacle
problem. As a result, under a superharmonicity condition on the obstacle, we
prove that each stratum of the singular set is locally contained in a single
manifold, up to a lower-dimensional subset, and the top stratum is locally contained in a
manifold
for some
if
.
In studying the top stratum, we discover a dichotomy, until now unseen, in this
problem (or, equivalently, the fractional obstacle problem). We find that second blow-ups
at singular points in the top stratum are global, homogeneous solutions to a codimension-2
lower-dimensional obstacle problem (or fractional thin obstacle problem) when
, whereas
second blow-ups at singular points in the top stratum are global, homogeneous, and
-harmonic
polynomials when
.
To do so, we establish regularity results for this codimension-2 problem, which we
call the very thin obstacle problem.
Our methods extend to the majority of the singular set even when no sign
assumption on the Laplacian of the obstacle is made. In this general case, we
are able to prove that the singular set can be covered by countably many
manifolds, up to a lower-dimensional subset.
