• Source: P-Laplacian

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


{\displaystyle p}

is allowed to range over



1
<
p
<
∞


{\displaystyle 1<p<\infty }

. It is written as





Δ

p


u
:=
d
i
v
(

|

∇
u


|


p
−
2


∇
u
)
.


{\displaystyle \Delta _{p}u:=div(|\nabla u|^{p-2}\nabla u).}


Where the




|

∇
u


|


p
−
2




{\displaystyle |\nabla u|^{p-2}}

is defined as






|

∇
u


|


p
−
2


=


[



(



∂
u


∂

x

1





)


2


+
⋯
+


(



∂
u


∂

x

n





)


2



]




p
−
2

2





{\displaystyle \quad |\nabla u|^{p-2}=\left[\textstyle \left({\frac {\partial u}{\partial x_{1}}}\right)^{2}+\cdots +\left({\frac {\partial u}{\partial x_{n}}}\right)^{2}\right]^{\frac {p-2}{2}}}


In the special case when



p
=
2


{\displaystyle 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
)


{\displaystyle W^{1,p}(\Omega )}

is a weak solution of





Δ

p


u
=
0


in


Ω


{\displaystyle \Delta _{p}u=0{\mbox{ in }}\Omega }


if for every test function



φ
∈

C

0


∞


(
Ω
)


{\displaystyle \varphi \in C_{0}^{\infty }(\Omega )}

we have





∫

Ω



|

∇
u


|


p
−
2


∇
u
⋅
∇
φ

d
x
=
0


{\displaystyle \int _{\Omega }|\nabla u|^{p-2}\nabla u\cdot \nabla \varphi \,dx=0}


where



⋅


{\displaystyle \cdot }

denotes the standard scalar product.




Energy formulation


The weak solution of the p-Laplace equation with Dirichlet boundary conditions






{



−

Δ

p


u
=
f




in


Ω




u
=
g




on


∂
Ω








{\displaystyle {\begin{cases}-\Delta _{p}u=f&{\mbox{ in }}\Omega \\u=g&{\mbox{ on }}\partial \Omega \end{cases}}}


in a domain



Ω
⊂


R


N




{\displaystyle \Omega \subset \mathbb {R} ^{N}}

is the minimizer of the energy functional




J
(
u
)
=


1
p




∫

Ω



|

∇
u


|


p



d
x
−

∫

Ω


f

u

d
x


{\displaystyle J(u)={\frac {1}{p}}\,\int _{\Omega }|\nabla u|^{p}\,dx-\int _{\Omega }f\,u\,dx}


among all functions in the Sobolev space




W

1
,
p


(
Ω
)


{\displaystyle W^{1,p}(\Omega )}

satisfying the boundary conditions in the trace sense. In the particular case



f
=
1
,
g
=
0


{\displaystyle f=1,g=0}

and



Ω


{\displaystyle \Omega }

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





)



{\displaystyle u(x)=C\,\left(1-|x|^{\frac {p}{p-1}}\right)}


where



C


{\displaystyle C}

is a suitable constant depending on the dimension



N


{\displaystyle N}

and on



p


{\displaystyle p}

only. Observe that for



p
>
2


{\displaystyle p>2}

the solution is not twice differentiable in classical sense.




See also


Infinity Laplacian




Notes






Sources


Evans, Lawrence C. (1982). "A New Proof of Local




C

1
,
α




{\displaystyle C^{1,\alpha }}

Regularity for Solutions of Certain Degenerate Elliptic P.D.E." Journal of Differential Equations. 45: 356–373. doi:10.1016/0022-0396(82)90033-x. MR 0672713.
Lewis, John L. (1977). "Capacitary functions in convex rings". Archive for Rational Mechanics and Analysis. 66 (3): 201–224. Bibcode:1977ArRMA..66..201L. doi:10.1007/bf00250671. MR 0477094. S2CID 120469946.




Further reading


Ladyženskaja, O. A.; Solonnikov, V. A.; Ural'ceva, N. N. (1968), Linear and quasi-linear equations of parabolic type, Translations of Mathematical Monographs, vol. 23, Providence, RI: American Mathematical Society, pp. XI+648, ISBN 9780821886533, MR 0241821, Zbl 0174.15403.
Uhlenbeck, K. (1977). "Regularity for a class of non-linear elliptic systems". Acta Mathematica. 138: 219–240. doi:10.1007/bf02392316. MR 0474389.
Notes on the p-Laplace equation by Peter Lindqvist
Juan Manfredi, Strong comparison Principle for p-harmonic functions

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