In mathematics, the p-Laplacian, or the p-Laplace operator, is a quasilinear elliptic partial differential operator of 2nd order. It is a nonlinear generalization of the Laplace operator, where
p
is allowed to range over
1
<
p
<
∞
. It is written as
Δ
p
u
:=
∇
⋅
(
|
∇
u
|
p
−
2
∇
u
)
.
Where the
|
∇
⋅
|
p
−
2
is defined as
|
∇
u
|
p
−
2
=
[
(
∂
u
∂
x
1
)
2
+
⋯
+
(
∂
u
∂
x
n
)
2
]
p
−
2
2
In the special case when
p
=
2
, this operator reduces to the usual Laplacian. In general solutions of equations involving the p-Laplacian do not have second order derivatives in classical sense, thus solutions to these equations have to be understood as weak solutions. For example, we say that a function u belonging to the Sobolev space
W
1
,
p
(
Ω
)
is a weak solution of
Δ
p
u
=
0
in
Ω
if for every test function
φ
∈
C
0
∞
(
Ω
)
we have
∫
Ω
|
∇
u
|
p
−
2
∇
u
⋅
∇
φ
d
x
=
0
where
⋅
denotes the standard scalar product.
The weak solution of the p-Laplace equation with Dirichlet boundary conditions
{
−
Δ
p
u
=
f
in
Ω
u
=
g
on
∂
Ω
in a domain
Ω
⊂
R
N
is the minimizer of the energy functional
J
(
u
)
=
1
p
∫
Ω
|
∇
u
|
p
d
x
−
∫
Ω
f
u
d
x
among all functions in the Sobolev space
W
1
,
p
(
Ω
)
satisfying the boundary conditions in the trace sense. In the particular case
f
=
1
,
g
=
0
and
Ω
is a ball of radius 1, the weak solution of the problem above can be explicitly computed and is given by
u
(
x
)
=
C
(
1
−
|
x
|
p
p
−
1
)
where
C
is a suitable constant depending on the dimension
N
and on
p
only. Observe that for
p
>
2
the solution is not twice differentiable in classical sense.
