Inverse iteration for the Monge–Ampère eigenvalue problem
Authors: Abedin Farhan , Kitagawa Jun 
Published: May 2020

We present an iterative method based on repeatedly inverting the Monge–Ampère operator with Dirichlet boundary condition and prescribed right-hand side on a bounded, convex domain

Ω
⊂

R

n

\Omega \subset \mathbb {R}^{n}

. We prove that the iterates

u
k

u_k

generated by this method converge as

k
→
∞

k \to \infty

to a solution of the Monge–Ampère eigenvalue problem
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="StartLayout Enlarged left-brace 1st Row 1st Column normal d normal e normal t upper D squared u equals lamda Subscript upper M upper A Baseline left-parenthesis negative u right-parenthesis Superscript n Baseline 2nd Column a m p semicolon in normal upper Omega comma 2nd Row 1st Column u equals 0 2nd Column a m p semicolon on partial-differential normal upper Omega period EndLayout">

{

d
e
t

D
2

u
=

λ

M
A

(
−
u

)
n

a
m
p
;

in 
Ω
,

u
=
0

a
m
p
;

on 
∂
Ω
.

\begin{equation*} \begin {cases} \mathrm {det} D^2u = \lambda _{MA} (-u)^n & \quad \text {in } \Omega ,\\ u = 0 & \quad \text {on } \partial \Omega . \end{cases} \end{equation*}

Since the solutions of this problem are unique up to a positive multiplicative constant, the normalized iterates
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="ModifyingAbove u With caret Subscript k Baseline colon-equal StartFraction u Subscript k Baseline Over StartAbsoluteValue EndAbsoluteValue u Subscript k Baseline StartAbsoluteValue EndAbsoluteValue Subscript upper L Sub Superscript normal infinity Subscript left-parenthesis normal upper Omega right-parenthesis Baseline EndFraction">

u
^

k

≔

u
k

|

|

u
k

|

|

L

∞

(
Ω
)

\hat {u}_k \coloneq \frac {u_k}{||u_k||_{L^{\infty }(\Omega )}}

converge to the eigenfunction of unit height. In addition, we show that
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="limit Underscript k right-arrow normal infinity Endscripts upper R left-parenthesis u Subscript k Baseline right-parenthesis equals limit Underscript k right-arrow normal infinity Endscripts upper R left-parenthesis ModifyingAbove u With caret Subscript k Baseline right-parenthesis equals lamda Subscript upper M upper A">

lim

k
→
∞

R
(

u
k

)
=

lim

k
→
∞

R
(

u
^

k

)
=

λ

M
A

\lim _{k \to \infty } R(u_k) = \lim _{k \to \infty } R(\hat {u}_k) = \lambda _{MA}

, where the Rayleigh quotient

R
(
u
)

R(u)

is defined as
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper R left-parenthesis u right-parenthesis colon-equal StartFraction integral Underscript normal upper Omega Endscripts left-parenthesis negative u right-parenthesis normal d normal e normal t upper D squared u Over integral Underscript normal upper Omega Endscripts left-parenthesis negative u right-parenthesis Superscript n plus 1 Baseline EndFraction period">

R
(
u
)

≔

∫

Ω

(
−
u
)
 

d
e
t

D
2

u

∫

Ω

(
−
u

)

n
+
1

.

\begin{equation*} R(u) \coloneq \frac {\int _{\Omega } (-u) \ \mathrm {det}D^2u}{\int _{\Omega } (-u)^{n+1}}. \end{equation*}

Our method converges for a wide class of initial choices

u
0

u_0

that can be constructed explicitly, and does not rely on prior knowledge of the Monge–Ampère eigenvalue

λ

M
A

\lambda _{MA}

.