Let
u
u
be a solution to the normalized
p
p
-harmonic obstacle problem with
p
>
2
p>2
. That is,
u
∈
W
1
,
p
(
B
1
(
0
)
)
u\in W^{1,p}(B_1(0))
,
2
>
p
>
∞
2>p>\infty
,
u
≥
0
u\ge 0
and
d
i
v
(
|
∇
u
|
p
−
2
∇
u
)
=
χ
{
u
>
0
}
in
B
1
(
0
)
\begin{equation*} \mathrm {div}( |\nabla u|^{p-2}\nabla u)=\chi _{\{u>0\}} \ \text { in } \ B_1(0) \end{equation*}
where
u
(
x
)
≥
0
u(x)\ge 0
and
χ
A
\chi _A
is the characteristic function of the set
A
A
. The main result is that for almost every free boundary point with respect to the
(
n
−
1
)
(n-1)
-Hausdorff measure, there is a neighborhood where the free boundary is a
C
1
,
β
C^{1,\beta }
-graph. That is, for
H
n
−
1
\mathcal {H}^{n-1}
-a.e. point
x
0
∈
∂
{
u
>
0
}
∩
B
1
(
0
)
x^0\in \partial \{u>0\}\cap B_1(0)
there is an
r
>
0
r>0
such that
B
r
(
x
0
)
∩
∂
{
u
>
0
}
∈
C
1
,
β
B_r(x^0)\cap \partial \{u>0\}\in C^{1,\beta }
.