Infinity Laplacian

In mathematics, the infinity Laplace (or

L

∞

{\displaystyle L^{\infty }}

-Laplace) operator is a 2nd-order partial differential operator, commonly abbreviated

Δ

∞

{\displaystyle \Delta _{\infty }}

. It is alternately defined, for a function

u

:

R

n

⟶

R

{\displaystyle u:\mathbb {R} ^{n}\longrightarrow \mathbb {R} }

of the variables

x

=

(

x

1

,

…

,

x

n

)

{\displaystyle x=(x_{1},\dots ,x_{n})}

, by

Δ

∞

u

=

⟨

D

u

,

D

2

u

D

u

⟩

=

∑

i

,

j

=

1

n

∂

2

u

∂

x

i

∂

x

j

∂

u

∂

x

i

∂

u

∂

x

j

{\displaystyle \Delta _{\infty }u=\langle Du,D^{2}u\,Du\rangle =\sum _{i,j=1}^{n}{\frac {\partial ^{2}u}{\partial x_{i}\,\partial x_{j}}}{\frac {\partial u}{\partial x_{i}}}{\frac {\partial u}{\partial x_{j}}}}

or

Δ

∞

u

=

⟨

D

u

,

D

2

u

D

u

⟩

|

D

u

|

2

=

1

|

D

u

|

2

∑

i

,

j

=

1

n

∂

2

u

∂

x

i

∂

x

j

∂

u

∂

x

i

∂

u

∂

x

j

,

{\displaystyle \Delta _{\infty }u={\frac {\langle Du,D^{2}u\,Du\rangle }{|Du|^{2}}}={\frac {1}{|Du|^{2}}}\sum _{i,j=1}^{n}{\frac {\partial ^{2}u}{\partial x_{i}\,\partial x_{j}}}{\frac {\partial u}{\partial x_{i}}}{\frac {\partial u}{\partial x_{j}}},}

where

D

u

{\displaystyle Du}

denotes the gradient vector,

D

2

u

{\displaystyle D^{2}u}

denotes the Hessian matrix, and

⟨

⋅

,

⋅

⟩

{\displaystyle \langle \cdot ,\cdot \rangle }

denotes the Euclidean inner product.

The first version avoids the singularity which occurs when the gradient vanishes, while the second version is homogeneous of order zero in the gradient. Verbally, the second version is the second derivative in the direction of the gradient. In the case of the infinity Laplace equation

Δ

∞

u

=

0

{\displaystyle \Delta _{\infty }u=0}

, the two definitions are equivalent.

While the equation involves second derivatives, usually (generalized) solutions are not twice differentiable, as evidenced by the well-known Aronsson solution

u

(

x

,

y

)

=

|

x

|

4

/

3

−

|

y

|

4

/

3

{\displaystyle u(x,y)=|x|^{4/3}-|y|^{4/3}}

. For this reason the correct notion of solutions is that given by the viscosity solutions.

Viscosity solutions to the equation

Δ

∞

u

=

0

{\displaystyle \Delta _{\infty }u=0}

are also known as infinity harmonic functions. This terminology arises from the fact that the infinity Laplace operator first arose in the study of absolute minimizers for

‖

D

u

‖

L

∞

{\displaystyle \|Du\|_{L^{\infty }}}

, and it can be viewed in a certain sense as the limit of the p-Laplacian as

p

→

∞

{\displaystyle p\rightarrow \infty }

. More recently, viscosity solutions to the infinity Laplace equation have been identified with the payoff functions from randomized tug-of-war games.