of 33
Proceedings of the Royal Society of Edinburgh
,
133A
, 773{805,
2003
Homogenization
of
a Hamilton{Jacobi
equation
associated
with
the
geometric
motion
of
an
interface
Bogdan
Craciun
Synopsys,
700 E. Middle­
eld Road,
Mountain
View,
CA 94043-4033,
USA
(
bogdan@synopsys.com
)
Kaushik
Bhattacharya
Division
of Engineering
and Applied
Science,
Mail
Stop
104-44,
California
Institute
of T echnology
, Pasadena,
CA 91125,
USA
(
bhatta@caltech.edu
)
(MS
received
17 January
2002;
accepted
16 January
2003)
This
paper
studies
the overall
evolution
of fronts
propagating
with
a normal
velocity
that
depends
on position,
v
n
=
f
(
x
), where
f
is rapidly
oscillating
and
periodic.
A
level-set
formulation
is used
to rewrite
this
problem
as the periodic
homogenization
of a Hamilton{Jaco
bi equation.
The
paper
presents
a series
of variational
characterization
(formulae)
of the e® ective
Hamiltonian
or e® ective
normal
velocity
.
It also
examines
the situation
when
f
changes
sign.
1. Introduction
The
present
paper
studies
the evolution
of interfaces
or fronts
propagating
with
a
normal
velocity
that
depends
on position.
Consider,
for example,
a free boundary
propagating
with
the normal
velocity
v
n
=
f
(
x
) =
¼
¡
~
¼
(
x
)
inside
a heterogeneous
body
occupying
a region
«
»
R
N
, where
¼
is a constant
and
~
¼
:
«
!
R
is a spatially
rapidly
oscillating
function.
This
problem
is motivated
by
the study
of phase
transformations
[4, 8], where
¼
is the applied
load
promoting
the
phase
transformation
and ~
¼
is the resistance
of the medium
to changing
phase.
If the
length-scale
of the oscillations
in ~
¼
are small
compared
to typical
length-scales
in
«
,
then
the detailed
evolution
of the front
is rather
complicated.
However,
it is often
possible
to de­ ne an overall
or e¬ ective
front
which
averages
over
the complicated
details.
This
paper
seeks
to ­ nd the law that
governs
the overall
evolution
of the
e¬ ective
front
when
«
=
R
N
and
~
¼
is periodic.
Similar
problems
arise
in the calculation
of ­ rst arrivals
in seismic
travel
times
(with
a velocity
that
varies
depending
on the type
of rock)
and
the development
process
in photolithography
, where
the resistive
strength
of the material
is di¬ er-
entially
altered
through
optical
processes
and
the material
is then
exposed
to an
etching
beam
that
removes
the weaker
material.
773
c
®
2003
The
Royal
Society
of Edinburgh
774
B.
Craciun
and
K.
Bhattacharya
An e ̄ cient
tool
for studying
such
problems
is the level-set
formulation.
If we
assume
that
there
exists
a smooth
function
h
:
R
N
£
[0
;
1
)
!
R
such
that
our
front
coincides
with
its zero
level
set at all times,
a simple
calculation
yields
n
=
r
h
jr
h
j
;
v
n
=
¡
h
t
jr
h
j
;
where
n
denotes
the normal
to the front
and
h
t
denotes
the derivative
of
h
with
respect
to the time
t
. It follows
that
h
satis­
es the following
Hamilton{Jacobi
initial-value
problem:
h
t
+
H
(
x;
r
h
) = 0
in
R
N
£
[0
;
1
)
;
h
(
x;
0) =
h
0
(
x
) in
R
N
;
with
the Hamiltonian
H
(
x; p
) =
f
(
x
)
j
p
j
(1.1)
and
the initial
data
h
0
2
BU C
(
R
N
) (where
BU C
denotes
the space
of bounded
and
uniformly
continuous
functions)
chosen
such
that
its zero
level
set coincides
with
the initial
position
of the front.
If
S
0
denotes
the set of all points
in the initial
front,
a common
choice
for
h
0
is
h
0
(
x
) = min(
d
(
x;
S
0
)
;
1)
;
where
d
(
x;
S
0
) is the distance
from
x
to
S
0
. This
level-set
approach
is reasonable
when
f
does
not change
sign,
since
the zero
level
set of the function
h
always
has
an empty
interior
by a result
of Barles
et al
. [2, theorem
4.1].
Now,
if the medium
in which
the front
is propagating
is periodic
with
unit
cell
[0
; "
]
N
, the corresponding
Hamilton{Jacobi
initial-value
problem
is
h
"
t
+
f
μ
x
"
jr
h
"
j
= 0
in
R
N
£
[0
;
1
)
;
h
"
(
x;
0) =
h
0
(
x
) in
R
N
;
9
>
=
>
;
(1.2)
with
f
continuous
and periodic
with
period
Y
N
= [0
;
1]
N
. Our
aim is to study
the
homogenization
of this
phenomenon,
i.e. to capture
its limit
behaviour
when
the
structure
of the medium
becomes
in­ nitely
­ ne (
"
!
0).
W e say that
a homogenized
front
exists
and its level-set
formulation
is given
by
the Hamilton{Jacobi
initial-v
alue
problem
h
t
+
·
H
(
r
h
) = 0
in
R
N
£
[0
;
1
)
;
h
(
x;
0) =
h
0
(
x
) in
R
N
)
(1.3)
if the viscosity
solution
h
"
of problem
(1.2)
converges
uniformly
on
R
N
£
[0
; T
) (for
all
T <
1
) to the viscosity
solution
of problem
(1.3).
W e call
·
H
the e¬ ective
or
homogenized
Hamiltonian.
F urthermore,
in analogy
to
H
(
x; p
) =
f
(
x
)
j
p
j
, we write
·
H
(
p
) = ·
v
n
(
p
)
j
p
j
Homogenization
of a Hamilton{Jacobi
equation
775
n
x
.
x
(
t
)
t
1
t
2
t
Figure
1. Snapshots
of the propagating
front
at times
t
1
,
t
and
t
2
,
and the trajectory
of a point
on the front.
and
call ·
v
n
the e¬ ective
normal
velocity
1
. Lions
et
al
. [12]
and
Ev ans [10]
have
shown
the existence
of the e¬ ective
Hamiltonian
under
certain
hypotheses,
which,
in our case,
reduce
to the strict
positivity
and Lipschitz
continuity
of
f
. They
also
discuss
some
properties
of the e¬ ective
Hamiltonian,
but their
results
do not provide
the means
for calculating
it.
The
present
paper
addresses
two issues:
­ rst the characterization
of the e¬ ective
Hamiltonian;
and
second
a discussion
of the situation
when
f
changes
sign.
Our
main
results,
theorems
3.4, 3.6 and 3.9, provide
a series
of variational
characteriza-
tions
or formulae
for
·
H
when
the Hamiltonian
is of the form
(1.1)
with
f
strictly
positive
(or equivalently
strictly
negative).
These
formulae
are based
on the Lax-
representation
formula
and the de­ nition
of the e¬ ective
Hamiltonian
in terms
of
this formula
by E [9]. In other
words,
the formulae
are based
on paths
and the time
to traverse/
distance
traversed
along
them.
Consider
a front
propagating
with
normal
velocity
v
n
=
f >
0 during
some
interv
al of time
(
t
1
; t
2
), as shown
in ­ gure
1. Let
x
(
t
) describe
the tra jectory
or path
followed
by a point
on this front.
Since
v
n
= _
x
¢
n
, it follows
that
j
_
x
j
(
t
)
>
f
(
x
(
t
)),
i.e. the speed
of the point
along
the path
is greater
than
or equal
to the normal
velocity
of the front.
Therefore,
the time
T
taken
to traverse
this same
path
with
speed
f
is greater
than
or equal
to the time
t
2
¡
t
1
taken
by the point
following
the
front,
T
=
Z
t
2
t
1
j
_
x
(
t
)
j
f
(
x
(
t
))
d
t
>
t
2
¡
t
1
:
It follows
that
the average
velocity
following
the front,
j
x
(
t
2
)
¡
x
(
t
1
)
j
=
(
t
2
¡
t
1
), is
greater
than
or equal
to the average
velocity
while
travelling
at speed
f
,
j
x
(
t
2
)
¡
x
(
t
1
)
j
=T
. F urthermore,
these
are equal
if and only
if the tra jectory
is always
normal
to the front.
Thus
it is intuitively
clear
that
the e¬ ective
velocity
can be obtained
as
the supremum
over
all paths
connecting
two traces
of the front.
Our
results
make
this intuition
precise.
Consider
any path
whose
endpoints
are separated
by the vector
d
. Let the time
to traverse
this path
with
speed
f
be
T
. Then
the pro jected
average
velocity
in the
1
We shall
see that
¹
v
n
(
p
) is homogeneous
of degree
zero
and
depends
only
on the direction
of
p
.
776
B.
Craciun
and
K.
Bhattacharya
direction
p
is
p
j
p
j
¢
d
T
:
Theorem
3.4 states
that
the e¬ ective
normal
velocity
·
v
n
(
p
) in the direction
p
is
obtained
by taking
the supremum
of this
pro jected
average
velocity
amongst
all
paths
of given
distance
j
d
j
=
D
and then
passing
to the limit
D
!1
. Theorem
3.6
states
that
the e¬ ective
normal
velocity
·
v
n
(
p
) in the direction
p
is obtained
by
taking
the supremum
of this pro jected
average
velocity
over
all paths
that
require
time
T
to traverse
and then
passing
to the limit
T
!1
. Finally
, theorem
3.9 states
that
the e¬ ective
normal
velocity
·
v
n
(
p
) in the direction
p
is obtained
by taking
the
supremum
of this pro jected
average
velocity
amongst
all paths
of given
pro jected
distance
p
¢
d
=
P
and passing
to the limit
P
!1
.
W e then
examine
the issues
that
arise
when
f
changes
sign
through
three
key
examples.
Example
5.1 shows
that
the front
may
assume
an oscillatory
shape
with
constant
amplitude
and frequency
of order
"
¡
1
as
"
decreases
2
. In short,
the front
converges
to a set with
an interior.
Example
5.2 shows
that
the front
may
become
trapped
in a smaller
and
smaller
set as
"
decreases
and
approaches
a stationary
front.
Finally
, example
5.3 shows
that
the front
may
shed
isolated
pieces
as it
propagates.
W e cannot
study
these
problems
with
the level-set
framework
discussed
above.
In fact,
we show
that
the homogenization
of (1.2)
makes
sense
only
when
f
assumes
strictly
positive
(or, equivalently
, strictly
negative)
values
(see remark
2.2).
Y et, in many
of these
examples,
one can identify
a clear
notion
of e¬ ective
front.
And
we ­ nd that
our variational
formulae
give
the velocity
of this front.
Since
these
formulae
are based
on the notion
of paths,
they
make
sense
even
when
f
changes
sign.
We conclude
that
when
f
changes
sign,
the e¬ ective
front
is decided
by the
nature
of paths
in regions
where
f
is strictly
positive
or negative,
and speci­
cally
whether
the paths
can percolate
through
regions
where
f
is strictly
positive
or
negative.
The paper
is organized
as follows.
Section
2 collects
the results
on homogenization
and also
shows
the necessity
of strict
positivity
for using
the level-set
formulation
and homogenization.
Section
3 derives
the v ariational
characterization
of the e¬ ec-
tive
normal
velocity
. Section
4 uses
these
formulae
to obtain
some
useful
bounds.
The
discussion
of the situation
when
f
changes
sign
is contained
in
x
5. This
sec-
tion
also contains
other
examples
to illustrate
the usefulness
of our formulae.
They
also
demonstrate
that
the homogenization
of isotropic
media
may
give
birth
to
anisotropic
ones.
2. Homogenization
We review
the literature
on the homogenization
of (1.2)
in this
section.
W e begin
with
a formal
asymptotic
expansion
(see
[3] for a systematic
presentation
of such
ansatz)
by assuming
that
h
"
(
x; t
) =
h
0
(
x; t
) +
"h
1
μ
x
"
; t
+
o
(
"
)
:
2
This
is allowed
by the fact that
the present
model
allows
the front
to develop
in ̄ nite curvature.
A model
that
avoids
this
by adding
a multiple
of
"μ
(where
μ
is the curvature)
to the normal
velocity
law is studied
in [7].
Homogenization
of a Hamilton{Jacobi
equation
777
Plugging
this into
(1.2)
and collecting
terms
of order
0, we ­ nd
h
0
t
+
f
(
y
)
jr
h
0
+
r
y
h
1
j
= 0
;
where
y
=
x="
. This
is a partial
di¬ erential
equation
for the corrector
h
1
and
its
solvability
condition
provides
a constraint
between
the partial
deriv
atives
of the
average
h
0
,
h
0
t
+
·
H
(
r
h
0
) = 0
;
with
the e¬ ective
Hamiltonian
·
H
is (uniquely)
determined
by the condition
that
a
periodic
solution
v
of
H
(
y; p
+
r
y
v
) =
·
H
(
p
)
(2.1)
exists.
This
result
has been
made
rigorous
by Lions
et al
. [12] and by Evans
[10].
Theorem
2.1
(
cf. [10])
.
Let
H
2
C
(
R
N
£
R
N
)
be
periodic
in
x
and
satisfy
H
(
x; p
)
!1
as
j
p
j!1
uniformly
for
x
2
R
N
:
(2.2)
For
each
p
2
R
N
, there
exists
a unique
2
R
(
which
we
denote
by
·
H
(
p
))
such
that
there
exists
a periodic,
viscosity
solution
v
2
C
(
R
N
)
of
H
(
y; p
+
D
y
v
) =
in
R
N
:
F urthermore,
·
H
is continuous
in
p
.
For
any
u
0
2
BU
C
(
R
N
) (
the
space
of bounded
and
uniformly
continuous
func-
tions
on
R
N
)
, the
solution
u
"
of the
Hamilton{Jacobi
equation
@u
"
@t
+
H
μ
x
"
; Du
"
= 0
in
R
N
£
[0
;
1
)
;
u
"
(
x;
0) =
u
0
(
x
)
in
R
N
converges
uniformly
on
R
N
£
[0
; T
] (
8
T <
1
)
to the
viscosity
solution
of
@u
@t
+
·
H
(
Du
) = 0
in
R
N
£
[0
;
1
)
;
u
(
x;
0) =
u
0
(
x
)
in
R
N
in
BU C
(
R
N
£
[0
; T
])
.
Moreover,
they
state
that
·
H
is a continuous
convex
function
and that
·
H
(
p
) goes
uniformly
to in­ nity
as
j
p
j!1
. They
also
derive
some
elementary
bounds,
inf
x
f
(
x
)
j
p
j
6
·
H
(
p
)
6
sup
x
f
(
x
)
j
p
j
:
(2.3)
While
this proves
the existence
of
·
H
, it does
not provide
a useful
characterization.
This
can be obtained
by looking
at its dual.
De­ ne the Lagrangian
associated
to
H
by its Legendre
dual
L
(
x; q
) = sup
p
2
R
N
(
q
¢
p
¡
H
(
x; p
)) =
(
0
if
j
q
j
6
f
(
x
)
;
1
if
j
q
j
> f
(
x
)
:
(2.4)
778
B.
Craciun
and
K.
Bhattacharya
In [9], E uses
¡
-convergence
techniques
for the Lax-representation
formula
h
"
(
x; t
) = inf
y
μ
h
0
(
y
) + inf
¹
μ
Z
t
0
L
μ
¹
"
;
_
¹
d
½
̄
̄
̄
̄
¹
(0) =
y; ¹
(
t
) =
x; ¹
2
W
1
;
1
(0
; t
)
and proves
that
the solution
of problem
(1.2)
converges
uniformly
to
inf
y
½
h
0
(
y
) +
t
·
L
μ
x
¡
y
t
¾
;
where
·
L
(
) = lim inf
D
!
1
1
D
inf
¿
2
H
1
0
(0
;D
)
Z
D
0
L
(
¶ t
+
¿
(
t
)
; ¶
+
_
¿
(
t
)) d
t;
(2.5)
and
that
·
L
is the dual
of
·
H
. This
result
thus
characterizes
·
H
using
a variational
principle.
Note
that
the hypothesis
(2.2)
in the theorem
is equivalent
to the strict
positivity
of
f
in our case,
i.e. when
H
is of the form
(1.1).
W e now
show
that
this is indeed
necessary
.
Remark
2.2
.
Assume
that
f
and
h
0
belong
to
W
1
;
1
(
R
N
). T o have
a non-station-
ary homogenized
level-set
function,
we need
f
to assume
strictly
positive
v alues
or
strictly
negative
values.
Proof.
We prove
this result
by contradiction.
Assume
that
the function
f
is neither
strictly
positive
nor strictly
negative.
By continuity
and
periodicity
of
f
, we can
­ nd a point
in any unit
cell where
it vanishes.
Since
the homogenized
front
does
move,
there
exists
some
compact
set
D
in
R
N
, with
non-void
interior,
in which
the function
h
changes
sign.
Without
loss of
generality
, we may
assume
that
h
0
(
x
)
>
0,
h
(
x; t
)
<
0 for all
x
in
D
and for some
strictly
positive
t
(note
that
we may
need
to change
f
to
¡
f
and
h
0
to
¡
h
0
for this
to be true).
By the uniform
convergence
of
h
"
to
h
, there
has to exist
a positive
"
D
such
that,
for any
"
smaller
than
"
D
,
h
"
(
x; t
)
<
0 in
D
.
On the other
hand,
since
the interior
of
D
is not void,
there
exists
some
positive
·
"
D
and some
x
D
in
D
such
that
x
D
+
"Y
n
»
D
8
"
6
·
"
D
.
Now
­ x
"
to be the minimum
of
"
D
and
·
"
D
. By our assumption
on
f
, we get a
point
x
2
D
in which
f
"
v anishes,
where
f
"
is de­ ned by
f
"
(
x
) =
f
μ
x
"
:
Due
to the uniform
continuity
of
h
"
, there
exists
a positive
real
m
such
that
h
"
(
x;
0)
> m;
h
"
(
x; t
)
<
¡
m
8
x
2
D:
(2.6)
Using
the existence
results
developed
by Lions
in [11,
ch. 9], we see that
the
viscosity
solution
of problem
(1.2)
belongs
to
W
1
;
1
(
R
N
£
(0
; t
]). Hence
its spatial
gradient
is bounded,
jr
h
"
j
6
M
a.e. in
D
£
(0
; t
]
;
(2.7)
Homogenization
of a Hamilton{Jacobi
equation
779
for some
positive
real
M
. Finally
, using
the continuity
of
f
and
the fact
that
it
vanishes
in
x
, we obtain
a ball
B
»
(
x
) contained
in
D
for which
f
(
y
)
<
2
m
tM
8
y
2
B
»
(
x
)
:
(2.8)
Now,
inequalities
(2.6)
give
us that
Z
t
0
h
t
(
z; ³
) d
³ <
¡
2
m
8
z
2
B
»
(
x
)
;
which
implies
Z
t
0
f
(
z
)
jr
h
"
j
d
³ >
2
m
8
z
2
B
»
(
x
)
:
Combining
this with
(2.7)
and (2.8),
we obtain
Z
t
0
2
m
tM
M
d
³ >
2
m
8
z
2
B
»
(
x
)
;
an obvious
contradiction,
since
Z
t
0
2
m
tM
M
d
³
= 2
m:
W e also note
that
E’s method
using
the Lax-representation
formula
also requires
the strict
positivity
of
f
, since
the Legendre
dual
is identically
in­ nity
otherwise.
Therefore,
we assume
in the next
section
that
f
is strictly
positive
everywhere
(if
f
is strictly
negative
everywhere,
change
f
to
¡
f
).
3. Characterization
of the e® ective
normal
velocity
We now
use the results
described
above
to ­ nd useful
characterizations
of the
homogenized
Hamiltonian
in the problem
motiv
ated
by phase
boundary
propaga-
tion
where
H
(
x; p
) =
f
(
x
)
j
p
j
:
We set
·
H
(
p
) = ·
v
n
(
p
)
j
p
j
(3.1)
and call ·
v
n
the e¬ ective
normal
velocity
.
Lemma
3.1
.
If
f
is Lipschitz
continuous,
Y
-periodic
and
f >
0
, the
e® ective
Hamil-
tonian
is given
by
·
H
(
p
) = sup
μ
cos(
¶ ; p
) lim sup
D
!
1
D
inf
®
2
D
0
;¶
D
T
(
®
)
j
p
j
;
(3.2)
where
cos(
a; b
) =
a
¢
b
j
a
jj
b
j
780
B.
Craciun
and
K.
Bhattacharya
denotes
the
cosine
of the
angle
formed
by
the
vectors
a
and
b
,
D
x;¶
D
is the
set
of al l
H
1
paths
connecting
x
with
x
+ (
¶ =
j
j
)
D
and
T
(
®
) =
Z
®
d
l
f
(d
l
is the
arclength
)
.
F urthermore,
·
v
n
is homogeneous
of degree
zero,
i.e.
it is even
and
depends
only
on
the
direction
p=
j
p
j
.
Proof.
If
v
is a viscosity
solution
corresponding
to
p
in (2.1),
then,
for any
¶ >
0,
notice
that
¶ v
is a solution
corresponding
to
¶ p
and
·
H
(
¶ p
) =
·
H
(
p
),
H
(
y; ¶ p
+
r
y
v
) =
f
(
y
)
j
¶ p
+
r
y
v
j
=
¶ f
(
y
)
j
p
+
r
y
v
j
=
¶ H
(
y; p
+
r
y
v
) =
·
H
(
p
)
:
By the uniqueness
of the cell problem,
it follows
that
·
v
n
is a function
of direction
only ,
·
v
n
(
p
) = ·
v
n
μ
p
j
p
j
:
Moreover,
proposition
2 in [12] ensures
that
·
v
n
is even.
Using
this property
of
·
H
, we get a ­ rst formula
for its Legendre
dual,
·
L
(
) = sup
p
(
¢
p
¡
·
H
(
p
)) = sup
p
j
p
j
(
j
j
cos(
¶ ; p
)
¡
·
v
n
(
p
)) =
(
0
if
j
j
6
F
(
)
;
1
if
j
j
> F
(
)
;
(3.3)
where
F
is de­ ned by
F
(
) = inf
p
½
·
v
n
(
p
)
cos(
¶ ; p
)
: cos(
¶ ; p
)
>
0
¾
:
F urthermore,
since
·
H
is continuous
and
convex,
it coincides
with
the dual
of its
dual,
so
·
H
(
p
) = sup
(
¢
p
¡
·
L
(
)) = sup
j
j
6
F
(
)
j
j
cos(
¶ ; p
)
j
p
j
:
Comparing
this with
(3.1),
we infer
·
v
n
(
p
) = sup
j
j
6
F
(
)
j
j
cos(
¶ ; p
) = sup
(
F
(
) cos(
¶ ; p
))
:
(3.4)
An alternative
formula
for
·
L
may
be obtained
from
E’s relation
(2.5).
Using
(2.4),
·
L
(
) =
(
0
if
9
D
n
!1
;
9
¿
n
2
H
1
0
(0
; D
n
)
3
:
j
+
_
¿
n
j
6
f
(
¶ t
+
¿
n
)
;
1
otherwise
:
(3.5)
Now
consider
some
with
j
j
> F
(
). By (3.3),
·
L
(
) =
1
, so (3.5)
implies
that,
for any
D
large
enough
and for any
¿
in
H
1
0
(0
; D
),
9
(
a; b
)
»
(0
; D
)
3
:
j
+
_
¿
(
t
)
j
> f
(
¶ t
+
¿
(
t
)) a.e. in (
a; b
)
:
(3.6)
Homogenization
of a Hamilton{Jacobi
equation
781
Having
D
and an arbitrary
¿
­ xed,
we pose
the following
initial-value
problem,
d
t
d
s
=
f
(
¶ t
+
¿
(
t
))
j
+
_
¿
(
t
)
j
;
t
(0) = 0
on the largest
interval
on which
j
+
_
¿
(
t
)
j
stays
strictly
positive
and
t
6
D
. Since
f
is periodic
and
continuous
and
¿
is in
H
1
0
(0
; D
), the derivative
of
t
is bounded
from
below
by a strictly
positive
number.
This
ensures
that
t
will eventually
get to
assume
the value
D
. Set
S
to be the corresponding
value
of
s
.
W e now
show
that
S > D
. Assume,
by contradiction,
that
S
6
D
. Then
we can
de­ ne a function
in
H
1
0
(0
; D
) by
Á
(
s
) =
(
¿
(
t
(
s
)) +
(
t
(
s
)
¡
s
) if
s
6
S;
(min
x
f
(
x
)
¡
1)
s
if
S < s
6
D;
which
satis­
es
̄
̄
̄
̄
+
d
Á
d
s
̄
̄
̄
̄
6
f
(
¶ s
+
Á
(
t
(
s
)))
a.e. in [0
; D
]
:
(3.7)
Since
(3.6)
and (3.7)
contradict
each
other,
it must
be that
S > D
, which
yields
D <
Z
S
0
d
s
=
Z
D
0
d
s
d
t
d
t
=
Z
D
0
j
+
_
¿
(
t
)
j
f
(
¶ t
+
¿
(
t
))
d
t:
What
we have
proved
up to now
is the following:
for any
with
j
j
> F
(
), any
su ̄ ciently
large
D
and any
¿
in
H
1
0
(0
; D
), we have
j
j
>
j
j
D
μ
Z
D
0
j
+
_
¿
(
t
)
j
f
(
¶ t
+
¿
(
t
))
d
t
¡
1
:
(3.8)
Equivalently
,
F
(
)
>
lim sup
D
!
1
sup
¿
2
H
1
0
(0
;D
)
j
j
D
μ
Z
D
0
j
+
_
¿
(
t
)
j
f
(
¶ t
+
¿
(
t
))
d
t
¡
1
:
(3.9)
W e now
aim for the converse
inequality
. Attempting
the same
trail,
consider
some
with
j
j
< F
(
). By (3.3),
·
L
(
) = 0, so (3.5)
implies
that
there
exists
a sequence
D
n
!1
and
¿
n
2
H
1
0
(0
; D
n
) such
that
j
+
_
¿
n
(
t
)
j
6
f
(
¶ t
+
¿
n
(
t
)) a.e. in [0
; D
n
].
This
yields
Z
D
n
0
j
+
_
¿
n
(
t
)
j
f
(
¶ t
+
¿
n
(
t
))
d
t
6
D
n
;
and hence
sup
¿
2
H
1
0
(0
;D
n
)
j
j
D
n
μ
Z
D
n
0
j
+
_
¿
n
(
t
)
j
f
(
¶ t
+
¿
n
(
t
))
d
t
¡
1
>
j
j
:
Thus
j
j
6
lim sup
D
!
1
sup
¿
2
H
1
0
(0
;D
)
j
j
D
μ
Z
D
0
j
+
_
¿
(
t
)
j
f
(
¶ t
+
¿
(
t
))
d
t
¡
1
8
with
j
j
< F
(
)
;
782
B.
Craciun
and
K.
Bhattacharya
which
gives
the converse
inequality
of (3.9).
W e conclude
that
F
(
) = lim sup
D
!
1
sup
¿
2
H
1
0
(0
;D
)
j
j
D
μ
Z
D
0
j
+
_
¿
(
t
)
j
f
(
¶ t
+
¿
(
t
))
d
t
¡
1
:
Since,
for any
¿
2
H
1
0
(0
; D
), the function
¶ t
+
¿
(
t
),
t
2
(0
; D
), describes
a
H
1
path
connecting
0 with
0 + (
¶ =
j
j
)
D
. W e can rewrite
this as
F
(
) = lim sup
D
!
1
μ
1
D
inf
®
2
D
0
;¶
D
Z
®
d
l
f
¡
1
;
(3.10)
and using
(3.4)
we reach
(3.2).
This
formula
for ·
v
n
has an interesting
interpretation.
According
to (3.10),
F
(
)
is the fastest
average
velocity
with
which
we can travel
a long
distance
in the
direction.
Equation
(3.2)
then
picks
the direction
such
that
the pro jected
velocity
in the direction
p
is the fastest.
Therefore,
·
v
n
is the fastest
pro jected
average
velocity
in the direction
p
. In particular,
a front
need
not directly
develop
with
normal
p
.
If a quicker
neighbouring
direction
exists,
the front
will rather
choose
to develop
portions
with
the corresponding
normal
at the microscopical
level
to obtain
a faster
propagation.
Lemma
3.2
.
The
function
G
(
x; y
) =
inf
¿
2
H
1
0
(0
;
1)
Z
1
0
j
y
¡
x
+
_
¿
(
t
)
j
f
(
x
+
t
(
y
¡
x
) +
¿
(
t
))
d
t
satis ̄
es
the
Lipschitz
continuity
condition
j
G
(
x
1
; y
1
)
¡
G
(
x
2
; y
2
)
j
6
j
x
1
¡
x
2
j
+
j
y
1
¡
y
2
j
inf
x
f
(
x
)
:
Proof.
Given
¿
1
2
H
1
0
(0
;
1), de­ ne
¿
2
2
H
1
0
(0
;
1) by
x
2
+
t
(
y
2
¡
x
2
) +
¿
2
(
t
) =
8
>
<
>
:
x
2
+ 3
t
(
x
1
¡
x
2
)
if
t
2
[0
;
1
3
]
;
x
1
+ (3
t
¡
1)(
y
1
¡
x
1
) +
¿
1
(3
t
¡
1) if
t
2
[
1
3
;
2
3
]
;
y
1
+ (3
t
¡
2)(
y
2
¡
y
1
)
if
t
2
[
2
3
;
1]
:
We have
that
G
(
x
2
; y
2
)
6
Z
1
0
j
y
2
¡
x
2
+
_
¿
2
(
t
)
j
f
(
x
2
+
t
(
y
2
¡
x
2
) +
¿
2
(
t
))
d
t
=
Z
1
=
3
0
3
j
x
1
¡
x
2
j
f
(
x
2
+ 3
t
(
x
1
¡
x
2
))
d
t
+
Z
2
=
3
1
=
3
3
j
y
1
¡
x
1
+
_
¿
1
(3
t
¡
1)
j
f
(
x
1
+ (3
t
¡
1)(
y
1
¡
x
1
) +
¿
1
(3
t
¡
1))
d
t
+
Z
1
2
=
3
3
j
y
2
¡
y
1
j
f
(
y
1
+ (3
t
¡
2)(
y
2
¡
y
1
))
d
t
Homogenization
of a Hamilton{Jacobi
equation
783
6
j
x
1
¡
x
2
j
min
x
f
+
Z
1
0
j
y
1
¡
x
1
+
_
¿
1
(
t
)
j
f
(
x
1
+
t
(
y
1
¡
x
1
) +
¿
1
(
t
))
d
t
+
j
y
1
¡
y
2
j
min
x
f
:
Since
this inequality
is satis­
ed for any
¿
1
2
H
1
0
(0
;
1), we have
veri­
ed that
G
(
x
2
; y
2
)
¡
G
(
x
1
; y
1
)
6
j
x
1
¡
x
2
j
min
x
f
+
j
y
1
¡
y
2
j
min
x
f
:
By symmetry
of
x
,
y
, we reach
the conclusion
of the lemma.
Lemma
3.3
.
(i)
The
starting
point
of the
curves
in
the
family
D
0
;¶
D
in
(3.2)
is arbitrary,
F
(
) = lim sup
D
!
1
D
inf
®
2
D
0
;¶
D
T
(
®
)
= lim sup
D
!
1
D
inf
®
2
D
D
T
(
®
)
;
where
D
D
is the
set
of all
H
1
paths
connecting
x
with
x
+ (
¶ =
j
j
)
D
for
some
x
2
R
N
.
(ii)
The
lim sup
in
(3.2)
is actual
ly a limit.
Proof.
(i) By considering
the [0
;
1] parametrizations
of the paths
in
D
0
;¶
D
, we get
inf
®
2
D
0
;¶
D
T
(
®
) =
inf
¿
2
H
1
0
(0
;
1)
Z
1
0
j
(
¶ =
j
j
)
D
+
_
¿
(
t
)
j
f
((
¶ =
j
j
)
D
+
¿
(
t
))
d
t
=
G
μ
0
;
j
j
D
:
Then
lemma
3.2 yields
1
D
inf
®
2
D
x;¶
D
T
(
®
)
6
1
D
inf
®
2
D
0
;¶
D
T
(
®
) +
2
j
x
j
D
min
y
f
(
y
)
8
D >
0
8
x
2
R
N
:
(3.11)
By the de­ nition
of lim sup,
there
exists
a sequence
D
n
!1
such
that
lim sup
D
!
1
D
inf
®
2
D
0
;¶
D
T
(
®
)
= lim
n
!
1
D
n
inf
®
2
D
0
;¶
D
n
T
(
®
)
:
Since
2
j
x
j
D
min
y
f
(
y
)
!
0 as
D
!1
;
equation
(3.11)
implies
lim sup
D
!
1
D
inf
®
2
D
0
;¶
D
T
(
®
)
6
lim
n
!
1
D
n
inf
®
2
D
x;¶
D
n
T
(
®
)
6
lim sup
D
!
1
D
inf
®
2
D
x;¶
D
T
(
®
)
8
x
2
R
N
:
The
converse
inequality
follows
in a similar
manner,
so we have
proved
lim sup
D
!
1
D
inf
®
2
D
0
;¶
D
T
(
®
)
= lim sup
D
!
1
D
inf
®
2
D
x;¶
D
T
(
®
)
8
x
2
R
N
:
from
which
we reach
our assertion.
784
B.
Craciun
and
K.
Bhattacharya
(ii) Let
¹
=
¶ =
j
j
and
L
= lim inf
D
!
1
1
D
inf
®
2
D
0
;¶
D
T
(
®
) = lim inf
D
!
1
G
(0
; D¹
)
D
:
Fix some
" >
0. By the de­ nition
of lim inf , there
exists
some
d
"
>
3
p
N
"
min
x
f
(
x
)
for which
̄
̄
̄
̄
L
¡
G
(0
; d
"
¹
)
d
"
̄
̄
̄
̄
6
1
3
":
Pick
some
integer
k
with
k
>
3
"
min
x
f
(
x
)
:
Let
D
"
=
kd
"
and choose
some
arbitrary
D
with
D > D
"
.
Now
pick
an integer
m
, with
m
>
k
, for which
D
2
(
md
"
;
(
m
+ 1)
d
"
]. Then
lemma
3.2 implies
G
(0
; D¹
)
6
G
(0
; md
"
¹
) +
D
¡
md
"
min
x
f
(
x
)
6
G
(0
; md
"
¹
) +
md
"
min
x
f
(
x
)
:
Since
m
>
k
and
k
>
3
=
(
"
min
x
f
(
x
)), this yields
G
(0
; D¹
)
D
6
G
(0
; D¹
)
md
"
6
G
(0
; md
"
¹
)
md
"
+
1
m
min
x
f
(
x
)
6
G
(0
; md
"
¹
)
md
"
+
1
k
min
x
f
(
x
)
6
G
(0
; md
"
¹
)
md
"
+
1
3
":
(3.12)
On the other
hand,
we clearly
have
G
(0
; md
"
¹
)
6
G
(0
;
(
m
¡
1)
d
"
¹
) +
G
((
m
¡
1)
d
"
¹ ; md
"
¹
)
:
(3.13)
Since
(
m
¡
1)
d
"
2
R
N
, there
exists
some
y
2
Z
N
such
that
j
y
¡
(
m
¡
1)
d
"
j
6
1
2
p
N
.
Due
to the periodicity
of
f
, we have
that
G
(
y; y
+
d
"
¹
) =
G
(0
; d
"
¹
)
:
By lemma
3.2 and this last equality
, equation
(3.13)
becomes
G
(0
; md
"
¹
)
6
G
(0
;
(
m
¡
1)
d
"
¹
) +
G
(
y; y
+
d
"
¹
) + 2
j
y
¡
(
m
¡
1)
d
"
j
min
x
f
(
x
)
6
G
(0
;
(
m
¡
1)
d
"
¹
) +
G
(0
; d
"
¹
) +
p
N
min
x
f
(
x
)
:
Homogenization
of a Hamilton{Jacobi
equation
785
Iterating
this inequality
, we obtain
G
(0
; md
"
¹
)
6
mG
(0
; d
"
¹
) + (
m
¡
1)
p
N
min
x
f
(
x
)
:
Recalling
the two assumptions
we made
on
d
"
,
G
(0
; md
"
¹
)
md
"
6
G
(0
; d
"
¹
)
d
"
+
m
¡
1
m
p
N
d
"
min
x
f
(
x
)
6
L
+
1
3
"
+
p
N
d
"
min
x
f
(
x
)
6
L
+
2
3
":
(3.14)
Adding
the inequalities
(3.12)
and (3.14),
we get
G
(0
; D¹
)
D
6
L
+
"
= lim inf
D
!
1
G
(0
; D¹
)
D
+
"
8
T > T
"
;
which
­ nally
allows
us to conclude
that
lim inf
D
!
1
G
(0
; D¹
)
D
=
L
= lim
D
!
1
G
(0
; D¹
)
D
:
This
equality
is su ̄ cient
to establish
our assertion.
W e now
present
our main
results.
Theorem
3.4
.
If
f
is Lipschitz
continuous,
Y
-periodic
and
f >
0
, the
e® ective
Hamiltonian
is given
by
·
H
(
p
) = sup
μ
j
p
j
cos(
¶ ; p
) lim
D
!
1
D
inf
®
2
D
D
T
(
®
)
= lim
D
!
1
sup
®
2
D
D
p
¢
d
(
®
)
T
(
®
)
;
(3.15)
where
d
(
®
)
is the
vector
joining
the
two
endpoints
of
®
and
D
D
is the
set
of
al l
H
1
paths
with
j
d
(
®
)
j
=
D
.
The
e® ective
Hamiltonian
is an
even
function
of
p
and
varies
linearly
with
its
length
j
p
j
.
Proof.
The
­ rst equality
is a direct
consequence
of the lemmas
above.
F or the
second,
we need
to prove
that
the limit
can be interchanged
with
the supremum.
Since
the expression
inside
the supremum
depends
only
on the direction
of
, it
is enough
to consider
only
the unit
vectors
(
2
@B
1
(0)).
Consider
the inequality
sup
2
@B
1
(
O
)
Dp
¢
inf
®
2
D
D
T
(
®
)
>
Dp
¢
inf
®
2
D
D
T
(
®
)
:
It follows
that
lim inf
D
!
1
sup
2
@B
1
(
O
)
Dp
¢
inf
®
2
D
D
T
(
®
)
>
lim
D
!
1
Dp
¢
inf
®
2
D
D
T
(
®
)
:
786
B.
Craciun
and
K.
Bhattacharya
since
the right-hand
side has a limit
by lemma
3.3. This
holds
for any
, so it follows
that
lim inf
D
!
1
sup
2
@B
1
(
O
)
Dp
¢
inf
®
2
D
D
T
(
®
)
>
sup
2
@B
1
(0)
lim
D
!
1
Dp
¢
inf
®
2
D
D
T
(
®
)
:
(3.16)
W e now
prove
the converse
inequality
. Consider
the expression
on the left-hand
side
of (3.16).
By the de­ nition
of lim sup,
we can ­ nd a sequence
D
n
! 1
such
that
lim inf
D
!
1
sup
2
@B
1
(
O
)
Dp
¢
inf
®
2
D
D
T
(
®
)
= lim
n
!
1
sup
2
@B
1
(
O
)
D
n
p
¢
inf
®
2
D
D
n
T
(
®
)
:
Since
@B
1
(0) is compact
and the expression
inside
the supremum
is continuous
with
respect
to
by lemma
3.2, the supremum
is attained
at
n
2
B
1
(0). W e thus
have
two sequences,
D
n
!1
and
f
n
g
n
2
N
2
@B
1
(0), such
that
lim sup
D
!
1
sup
2
@B
1
(0)
D¶
¢
p
inf
®
2
D
D
T
(
®
)
= lim
n
!
1
D
n
n
¢
inf
®
2
D
n
D
n
T
(
®
)
:
Using
again
the compactness
of
@B
1
(0), and restricting
to a subsequence
if neces-
sary , the sequence
n
converges
to some
2
@B
1
(0).
A direct
application
of lemma
3.2 gives
inf
®
2
D
n
D
n
T
(
®
) =
G
(0
; D
n
n
)
>
G
(0
; D
n
)
¡
D
n
j
¡
n
j
min
x
f
(
x
)
= inf
®
2
D
D
n
T
(
®
)
¡
D
n
j
¡
n
j
min
x
f
(
x
)
:
Thus
inf
®
2
D
n
D
n
T
(
®
)
D
n
>
inf
®
2
D
D
n
T
(
®
)
D
n
¡
j
¡
n
j
min
x
f
(
x
)
and,
since
n
¢
p
!
¢
p
as
n
!1
,
lim sup
D
!
1
sup
2
@B
1
(0)
D¶
¢
p
inf
®
2
D
D
T
(
®
)
= lim
n
!
1
D
n
n
¢
p
inf
®
2
D
n
D
n
T
(
®
)
6
lim
n
!
1
D
n
¢
p
inf
®
2
D
D
n
T
(
®
)
= lim
D
!
1
D¶
¢
p
inf
®
2
D
D
T
(
®
)
6
sup
2
@B
1
(0)
lim
D
!
1
D¶
¢
p
inf
®
2
D
D
T
(
®
)
:
This
inequality
and (3.16)
conclude
the proof.
Remark
3.5
.
The
e¬ ective
velocity
behaves
monotonically
with
respect
to
f
.