ondition and appliations to the mean urvature ow
in IR 2
Guy Barles, Samuel Biton and Olivier Ley
Laboratoire de Mathematiques et Physique Theorique
Universite de Tours
Par de Grandmont, 37200 Tours, Frane
Abstrat
In this artile, we prove a omparison result for visositysolutions of a ertain
lass of fully nonlinear, possibly degenerate, paraboli equations; the main new
featureofthisresultisthatitholdsforany,possiblydisontinuous,solutionswithout
imposinganyrestritionson their growth at innity. The mainappliation of this
resultwhihwasalsoourmainmotivationtoproveit, istheuniquenessofsolutions
to one-dimensional equations inluding the mean urvature equation for graphs
withoutassuminganyrestritionon theirbehaviorat innity.
Key-words: quasilinearparaboliequations,meanurvatureequations,uniquenesswith-
out growth onditions, visosity solutions.
AMS subjet lassiations: 35A05, 35B05,35D05,35K15, 35K55,53C44
1 Introdution
The main motivation of this paper omes from the following, rather surprising, result
of Eker and Huisken [13℄: for any initial data u
0 2 W
1;1
lo (IR
N
), there exists a smooth
solution ofthe equation
8
<
: u
t
u+
hD 2
uDu;Dui
1+jDuj 2
=0 in IR N
(0;+1);
u(x;0)=u
0
(x) inIR N
:
(1)
ThisworkwaspartiallysupportedbytheTMRprogram\Visositysolutionsandtheirappliations."
Here and belowthe solutionu isa real-valuedfuntion, Duand D u denote respetively
itsgradientand Hessian matrixwhile jjand h;istandfor the lassialEulideannorm
and inner produt inIR N
.
In order to be omplete, it is also worth mentioning that, following Angenent [1℄,
one an extend this result to merely ontinuous initial data using the interior gradient
estimates of Evans and Spruk [15℄ (see [6℄).
The very non-standard feature of this kind of results is that no assumption on the
behavior of the initial data at innity is imposed and therefore the solutions may also
haveany possible behavior atinnity. A naturaland intriguingquestion isthen whether
suh asolution isunique or not.
This question leads us to study of the uniqueness properties for unbounded visosity
solutionsofquasilinear,possiblydegenerate,paraboliequationssetinIR N
andthisartile
is aontinuation of this programstarted in[6℄ (see also[5℄, [4℄ and [7℄).
In this paper, we are more partiularlyinterested in the one-dimensional ase for (1)
where this equation redues to
8
<
: u
t u
xx
1+u 2
x
=0 inIR(0;T);
u(;0)=u
0
inIR :
(2)
One of the mainontributionof this work isa uniqueness result for smooth solutions
of (2) (and even for more generalequations), without any restrition on their growth at
innity (see Theorem4.2); itimmediatelyyieldsa omparisonresultfor, possibly dison-
tinuous, visosity solutions of (2)by the geometrial approah of [6℄ (f. Corollary 5.1),
i.e. a omplete answer to the question we address. After this work was ompleted, we
learn that Chou and Kwong [9℄ proved general uniqueness results for a ertain lass of
quasilinear paraboli equations in the one-dimensional ase: their hypothesis are of dif-
ferent nature from ours and their methods are quite dierent but their result, whih is
alsovalid for solutionswithoutany restrition on their behaviorat innity,inludes also
the mean urvature equation.
This kindofuniqueness resultsisnon-standard: itiswell-knownthatuniqueness fails,
in general, if we do not impose growth onditions on the solutions at innity, like, for
example,the heat equation. The only(general) ase wheresimilar resultshold isthe one
ofrst-orderequationswheretheyareaonsequeneof\nitespeedofpropagation"type
properties(see Crandalland Lions [12℄,Ishii[19℄ and Ley[20℄) butthey areveryunusual
for seond-order equation.
Our uniqueness proof for(2) onsistsin integrating the equation, whih leads toon-
sider the pde
v
t
artan(v
xx
)=0 in IR(0;+1): (3)
Andthisisthisnewequationwhihisshowntosatisfyaomparisonprinipleforvisosity
solutionswithout growth ondition atinnity.
and Capuzzo Doletta [2℄ or Barles [3℄, et. for a presentation of the notion of visosity
solutionsand forthe key basi results.
In fat, the omparison result we are going to prove and apply to (3) holds in IR N
(and not only indimension 1) and for moregeneral equations of the type
8
<
: u
t
+F(x;t;Du;D 2
u)=0 in IR N
(0;T);
u(;0)=u
0
in IR N
;
(4)
whereF isaontinuousfuntionfromIR N
[0;T℄IR N
S
N
intoIRandsatisessuitable
assumptions (see (H1) and (H2) in Setion 2). Our proof relies essentially on the use
of a \friendly giantmethod." This result implies the uniqueness for smooth solutions of
quite generalequations in the one-dimension spae inluding(2) as we mention itabove
(see Setion 4).
Finally we investigate the onsequenes of the existene and uniqueness of solutions
for (2) from the geometrial point of view. The geometrial interpretation of (2) is that
the graph of u atevery time t0,namely
t
:=Graph(u(;t));
is a smooth hypersurfae of IR 2
whih evolves by its mean urvature. Suh evolution
has been studiedreently by the so-alled\level-set approah": introduedfor numerial
purposes by Osher and Sethian [21℄, this approah was developed by Evans and Spruk
[14℄ and Chen, Giga and Goto [8℄; the generalized motion of hypersurfaes is desribed
through the evolution of the level-sets of solutions of a suitable geometrial pde. This
approah oersa lot of advantages: itis possible to deal with nonsmooth hypersurfaes,
itallows numerialomputations,et. Themain issueis theagreementwith the lassial
motiondenedindierentialgeometry. Thesequestionswereaddressedbyalotofauthors
(see espeiallyEvans and Spruk [14℄, [15℄ and Ilmanen [18℄).
Ouruniquenessresultallowsustoshowthat,intheaseofthemeanurvaturemotion
of any entire ontinuous graph in IR 2
, the level-set approah agrees with the lassial
motion dened in dierential geometry. As a by-produt, we have a omparison result
between possibly disontinuous visosity sub and supersolutions of (2) and therefore the
uniqueness holds even in the lass of disontinuous solutions. We refer to Setion 5 for
the omplete statement of these results.
The paper is divided as follows. In Setion 2, we prove the strong omparison prin-
iple for visosity solutions of equation (4). Setion (3) is devoted to the proof of the
existeneofasolutionfor(4). InSetion4,weapplythepreviousresulttouniquenessfor
someone-dimensional equationsinluding(2)asapartiularase. In thelastsetion,we
statesome geometrialonsequenesforthe meanurvatureowinIR 2
andmore general
geometrial motions.
dotoral position at the University of Padova. He would like to thank the department
of mathematis and espeially Martino Bardi for their kind hospitality and fruitful ex-
hanges.
2 A omparison result for visosity solutions without
growth onditions at innity
In order to state the omparison priniple for visosity solutions of equation (4), we use
the followingassumptions inwhih
B(x
0
;R )=fx2IR N
:jx x
0
jR gdenotes the ball
of radius R > 0 entered in x
0 2 IR
N
and S
N
the spae of N N symmetri matries;
moreover, for every symmetri matrix X = P(diag(
i )
1iN )P
T
; where the
i
's are the
eigenvalues of X and P is orthogonal, we deneX +
=P(diag(
+
i )
1iN )P
T
:
(H1) For any R>0; thereexists a funtionm
R :IR
+
!IR
+
suh that m
R (0
+
)=0 and
F(y;t;(x y);Y) F(x;t;(x y);X)m
R
jx yj 2
+jx yj
;
for allx;y2
B(0;R ); t2[0;T℄;X;Y 2S
N
; and >0suh that
3
I 0
0 I
X 0
0 Y
3
I I
I I
:
(H2) There exists 0<<1and onstantsK
1
>0 and K
2
>0 suh that
F(x;t;p;X) F(x;t;q;Y)K
1
jp qj(1+jxj)+K
2
(Tr[Y X℄
+
)
;
for every (x;t;p;X;Y)2IR N
IR N
[0;T℄S
N S
N :
Our result is the following
Theorem 2.1 Assume that (H1)and (H2) holdand let u (respetively v) be an upper-
semiontinuousvisositysubsolution (respetively lower-semiontinuousvisositysuperso-
lution) of (4). If u(;0)v(;0) in IR N
; then uv in IR N
[0;+1):
Before proving this result, letus ommentthe assumptions. Assumption (H1)is the
lassialone toensureuniqueness forthis typeof fullynonlinear paraboliequations(see
Condition(3.14)of [11℄): herewejust stateitinamoreloalway. The assumption(H2)
ontains allthe restrition whihallows suh a resultto hold: the rst term of the right-
handsideisthelassialonetoobtain\nitespeedofpropagation"typeresultswhilethe
seond one onerns the behavior of F in D 2
u, and in partiular at innity as (3) shows
it. Obviously suh an assumption or a related one is needed sine suh a result annot
be true forthe heat equation. A nonlinearityF whih satises(H2)is neessarilyof the
1 2
ellipti. We donot know if these assumptions are lose or far tobe optimal.
As an exampleof pde whihsatises (H1){(H2), we have
u
t
(a(x;t)u) +
+b(x;t)jDuj=f(x;t) in IR N
(0;T);
where a;b are ontinuous funtions onIR N
(0;T), Lipshitz ontinuous in x uniformly
with respet to t, f 2C(IR N
(0;T))and 0<<1.
Nowwe turn to the proof of the theorem.
Proof of Theorem 2.1. The proof is done in two steps: the rst one is a kind of
linearization proedure whihyields a new pde foru v:The seondone onsistsin the
onstrution of a suitable smooth supersolution of this new pde whih blows up at the
boundary of balls.
The rst step isdesribed in the
Lemma 2.1 Under the assumptions of Theorem 2.1, the upper-semiontinuousfuntion
!:=u v is a visosity subsolution of
A[!℄=
!
t K
1
(1+jxj)jD!j K
2
Tr(D 2
!) +
=0 in IR N
(0;+1): (5)
Moreover !(;0)0 in IR N
.
Then the seondstep relies on the
Lemma 2.2 There exists > 0 and k > > 0 suh that, for R > 0 large enough, the
smooth funtion
R
(x;t)='
(1+R 2
) 1
2
(1 t) (1+jxj 2
) 1
2
with '(r)=R
=r k
; (6)
is a strit supersolution of (5) in the domain
D(;R )=f(x;t)2IR N
[0;T℄: t<
1
2
; (1+jxj 2
) 1
2
<(1+R 2
) 1
2
(1 t)g: (7)
Wepostpone the proof of the lemmasand we rst onlude the proofof the theorem.
Weonsider sup
D(;R)
f!
R
g ; sine
R
goesto +1on the lateralside of D(;R ), this
supremum is ahieved atsome point(x;
t)2D(;R ).
We annot have
t> 0 beause, otherwise sine by Lemma 2.1! is a visosity subso-
lution of (5), we would have A[
R (x ;
t)℄ 0, whih would ontradit the fat that
R is
a strit supersolution of this pde by Lemma 2.2.
Thus, neessarily t = 0 and the maximum is nonpositive sine !(;0) 0 and
R
(;0)>0 inIR N
. It follows that,for every (x;t) 2D(;R );
(u v)(x;t)
R
[(1+R 2
) 1
2
(1 t) (1+jxj 2
) 1
2
℄ k
: (8)
Inorder toonlude,we letR goto+1inthisinequalityfor t<1=2: sine k > >0,
weobtain (u v)(x;t)0 forany x2IR N
. Toprovethe sameproperty forallt2[0;T℄,
we iteratein time and the proof of the theorem isomplete. 2
We turn tothe proof of the lemmas.
Proof of Lemma 2.1. Let 2 C 2
(IR N
(0;+1)) and suppose that (x ;
t) 2 IR N
(0;+1)is astrit loalmaximum pointof ! =u v and morepreisely astrit
maximum point of this funtion in
B(x;) 2
[
t ;
t+℄.
We onsider, for ">0,
max
B(x;) 2
[
t ;
t +℄
u(x;t) v(y;t) (x;t)
jx yj 2
"
2
: (9)
Bylassial arguments, itis easytoprovethat the maximum in(9)is ahieved atpoints
(x
"
;y
"
;t
"
) suh that
(x
"
;y
"
;t
"
)!(x ;x;
t) and jx
"
y
"
j 2
"
2
!0 as "!0: (10)
Hene, for " small enough, (x
"
;y
"
;t
"
) 2 B(x ;)B(x;)(
t ;
t+) and following
[11℄, there exista;b2IR ; X;Y 2S
N
; suh that,if weset p
"
:=
2(x
"
y
"
)
"
2
, we have
(a;D(x
"
;t
"
)+p
"
;X+D 2
(x
"
;t
"
))2
P 2;+
u(x
"
;t
"
);
(b;p
"
;Y)2
P 2;
v(x
"
;t
"
)
and suh that a b=
t (x
"
;t
"
)and
6
"
2
I 0
0 I
X 0
0 Y
6
"
2
I I
I I
:
Usingthat u and v are respetively visosity sub- and supersolution of (4), we have
t (x
"
;t
"
)+F x
"
;D(x
"
;t
"
)+p
"
;X+D 2
(x
"
;t
"
)
F(y
"
;p
"
;Y)0:
F (y
"
;p
"
;Y) F(x
"
;p
"
;X) m
R
2 jx
"
y
"
j 2
"
2
+jx
"
y
"
j
and from (H2),we get
F x
"
;D(x
"
;t
"
)+p
"
;X+D 2
(x
"
;t
"
)
F (x
"
;p
"
;X)
K
1 jD(x
"
;t
"
)j(1+jx
"
j) K
2
Tr(D 2
(x
"
;t
"
)) +
:
Finally,we obtain
t (x
"
;t
"
) K
1 jD(x
"
;t
"
)j(1+jx
"
j) K
2
Tr (D 2
(x
"
;t
"
)) +
m
R
2 jx
"
y
"
j 2
"
2
+jx
"
y
"
j
:
Letting " goto 0and using (10) yields
t (x;
t) K
1
jD(x;
t)j(1+jx j) K
2 Tr[D
2
(x;
t)℄
+
0
whih is exatlythe inequality showing that ! is a subsolution of (5). 2
Proof of Lemma 2.2. In order to hek that is a strit supersolution for a suitable
hoieofonstantsk > >0and>0independentofR ,wesetr =(1+R 2
) 1=2
(1 t)
(1+jxj 2
) 1=2
and note that, for (x;t) 2 D(;R ); we have r 2 [0;
R ℄ where
R = p
1+R 2
:
Moreover, the funtion 'is dereasing onvex, i.e.
' 0
<0; ' 00
0 on (0;
R℄:
The omputation ofthe derivativesof gives
t
= (1+R 2
) 1
2
' 0
; D= '
0
x
(1+jxj 2
) 1
2
;
D 2
= '
0
I
(1+jxj 2
) 1
2
xx
(1+jxj 2
) 3
2
!
+' 00
xx
1+jxj 2
:
Substituting the derivativesin A[℄(see Lemma2.1), we have
A[℄(x;t)= (1+R 2
) 1=2
' 0
(r) K
1
(1+jxj)j' 0
(r)j
jxj
(1+jxj 2
) 1=2
K
2 Tr
"
' 0
(r)
I
(1+jxj 2
) 1
2
xx
(1+jxj 2
) 3
2
!
+' 00
(r)
xx
1+jxj 2
#
+
!
(11)
We estimate(11) frombelow. First
K
1
(1+jxj)j' 0
(r)j
jxj
(1+jxj 2
) 1
2
2K
1 '
0
(r)
R; (12)
for every (x;t)2D(;R ) sine j' 0
(r)j= ' 0
(r)0:Notiing that the matries
X = ' 0
(r)
I
(1+jxj 2
) 1
2
xx
(1+jxj 2
) 3
2
!
and Y =' 00
(r)
xx
1+jxj 2
are nonnegative and, using the inequality (a+b)
a
+b
for a;b 0 and 0< <1;
we obtain
(Tr[X +Y℄ +
)
=(TrX +Tr Y)
(TrX)
+(TrY)
N
( ' 0
(r))
+N
(' 00
(r))
:(13)
From (11), (12) and (13), it follows
A[℄(x;t) ' 0
(r)( 2K
1 )
R K
2 N
' 0
(r)
K
2 N
' 00
(r)
k( 2K
1 )
R R
r k+1
K 0
2 R
r (k+1)
K 0
3 R
r (k+2)
R
r k+1
k( 2K
1 )
R K
0
2 R
( 1)
r
(k+1)(1 )
K 0
3 R
( 1)
r
(k+1) (k+2)
| {z }
S(r)
(14)
where K 0
2
and K 0
3
are positive onstants whih depend onlyon N;K
2
;k and :
We want to hoose ; and k inorder that the quantity S(r) in (14) is positive. We
rst hoose k suh that the quantity S(r) is noninreasingon [0;
R ℄ : sine the exponent
(k +1)(1 ) of the rst term in r is already positive for every hoie of k > 0, it is
enoughto take
k
2 1
1
=) (k+1) (k+2)0; (15)
inorder to ensurethat the exponentof the seondterm in r is alsopositive.
Wearethenlefttohoose parametersinordertohaveS(
R )>0;todoso,aneessary
ondition is learly
>2K
1
: (16)
Usingthe fatthat R p
1+R 2
=
R , wehave
S(
R ) = k( 2K
1 )
R K
0
2 R
( 1)
R
(1 )(k+1)
K 0
3 R
( 1)
R
(k+1) (k+2)
k( 2K
1 )
R K
0
2
R
(1 )(k+1 )
K 0
3
R
( 1)+(k+1) (k+2)
:
Now it is lear that, if we an take the exponents of R in the two last terms stritly
lessthan 1,thenweare donesineforlargeR ,theright-handsideofthe inequalitywould
be stritly positive.
This yieldsthe followingonditions
(1 )(k+1 )<1; (17)
( 1)+(k+1) (k+2)<1: (18)
We reall that 0<<1 ; onone hand, an easyomputation shows that Condition (18)
isautomatiallysatised when(17)holds. On theother hand,Condition (17)holds ifwe
hoose suh that
k >>k
1
: (19)
Finally,we an xall the onstantsin orderto fulll(15), (16)and (19) and thus allthe
required propertiesof are satised. 2
In fat, a lose look at the proof of Theorem 2.1 shows that the existene of the
\friendly giants"
R
allows to get some loal estimate on the dierene between two
solutions. It leads to the following kind of stability result whih willbe useful later.
Proposition 2.1 Let (F
"
)
">0
a family of ontinuous funtions satisfying assumptions
(H1)and(H2)uniformlywithrespetto". Assumethatthereexistsontinuousvisosity
solutionsu
"
andv
"
of equation(4) withF replaedbyF
"
suhthat(u
"
v
"
)(;0)onverges
loallyuniformlyto0in IR N
. Thenu
"
v
"
onvergesloallyuniformlyto0in IR N
[0;T℄.
ProofofProposition2.1. Sinethegeneralasewillfollowbyiteratingintime,weonly
prove the result for t 1=2 realling that is the onstant appearing in the denition
of the \friendlygiants"
R
(see (6)).
Werst observethat, forany R>0,thefuntion u
"
sup
B(0;R) f(u
"
v
"
)(;0)gisstill
a solutionof (4)with F replaed by F
"
and that we have
u
"
(;0) sup
B(0;R) f(u
"
v
"
)(;0)gv
"
(;0) in B(0;R ):
Sine F
"
satises (H2) independently of "; we get from inequality (8) that, for all
(x;t)2D(;R );
u
"
(x;t) v
"
(x;t) sup
B(0;R) f(u
"
v
"
)(;0)g+
R (x;t):
In order to prove that u
"
v
"
onverges loally uniformlyto 0, we take an arbitrary
>0and(x;t)inaompatsubsetK ofIR N
[0;1=2℄. ForR suÆientlylarge,wehave
R
(x;t)=2inK. Then,for"suÆientlysmall,wehavesup
B(0;R) f(u
"
v
"
)(;0)g=2:
Therefore u
"
(x;t) v
"
(x;t) inK. And thuslimsup
"!0 sup
K (u
"
v
"
)0.
By exhanging the roles of u
"
and v
"
, we obtain the lower estimate and the proof is
omplete. 2
In this setion we use the omparison theorem to get the followingexistene result.
Theorem 3.1 Suppose that (H1) and (H2) hold. Then, for every initial datum u
0 2
C(IR N
); there exists a unique ontinuous visosity solution of (4).
Proof of Theorem 3.1. The uniqueness part is given by Theorem 2.1. For the exis-
tene, we set '
"
(r) := minfmaxfr; 1="g;1="g for every r 2 IR and " > 0 and onsider
trunations of pde (4) of the form
u
t +F
"
(x;t;Du;D 2
u)=0; (20)
whereF
"
='
"
ÆF:Notiingthat'
"
(a) '
"
(b)maxfa b;0g;weseethat theF
"
satisfy
assumptions (H1)and (H2), uniformlywith respet to":
Werstget anexisteneresultfor(20)usingPerron'smethod. SinejF
"
(x;t;p;M)j
1=" for all (x;t;p;M); the funtion u
"
(x;t) := u
0
(x) + t=" (respetively u
"
(x;t) :=
u
0
(x) t=") is a supersolution (respetively a subsolution) of (20). Theorem 2.1 pro-
vides a strong omparison result for (20) and therefore Perron's method applies readily
givingthe existene of a ontinuous solutionu
"
of (20) with initialdatum u
0 .
The next step onsists in deriving a loal L 1
-bound for the family (u
"
)
">0
we built
above. To do so, we use of the \friendly giants" introdued in Setion 2. Let R ; > 0
and D(;R ) dened by formula(7). Weset
C
R
=2 sup
B(0;R) ju
0 j; K
R
=2 sup
D(;R)
jF(;;0;0)j2 sup
D(;R) jF
"
(;;0;0)j;
and we onsider the funtions (x;t) 7! C
R K
R
t
R
(x;t) and (x;t) 7!C
R +K
R t+
R
(x;t)inD(;R ). Tediousbutstraightforwardomputationsshows thatthesefuntions
are respetively sub- and supersolution of (20) in D(;R ) and it is lear that we have
C
R
R
(x;0)u
"
(x;0)C
R +
R
(x;0)inB(0;R ). Then,easyomparisonarguments
showthat
C
R K
R
t
R u
"
C
R +K
R t+
R
inD(;R ):
Itfollows thatthe family(u
"
)
">0
isbounded inD(;R =2)independently of": Iteratingin
time, we get the loalboundedness of (u
"
)
">0 inIR
N
[0;+1):
Finally we apply the \half-relaxed-limits"methodwhih onsistsin introduing
u=limsup
"!0 +
u
"
and u=liminf
"!0 +
u
"
whiharewell-denedbeauseoftheloalL 1
-boundontheu
"
's. Moreover,theyareboth
disontinuous solutionsof (4)withinitialdatumu
0
sine (F
"
) onvergesloallyuniformly
toF:The strong omparisonresult for(4) (Theorem 2.1) shows thatu=u:=uand uis
the desired ontinuous solutionof (4)we wanted tobuild. 2
In this setion, we provide some appliations of the previous result in the ase of one-
dimensional quasilinearparaboliequations. Weaddress here onlyuniqueness questions.
We onsider the equation
8
<
: u
t
(f(x;t;u;u
x ))
x
=0 inD 0
(IR(0;+1));
u(;0)=u
0
in IR ;
(21)
where u
0
2C(IR ); the nonlinearityf 2C(IR[0;+1)IRIR ) and D 0
(IR(0;+1))
is the spae of distributions on IR(0;+1): For reasons whih will be lear below, we
onsider only ases when the solution uis in C 1
(IR(0;+1)).
Tostate our result, we use the following assumption onf
(H3)f isloallyLipshitzontinuousinIR(0;+1)IRIRandthereexists onstants
C >0and 0<<1suh that, forany t>0and x;y;u;v;p;q2IR , we have
f(x;t;u;p) f(y;t;v;q)C
(1+jpj+jqj)jx yj+(1+jxj+jyj)ju vj+ (p q) +
:
Note that this assumption imply(H1)and (H2) inthe one-dimensional ase. Equa-
tion (21) makes sense in the spae of distributions sine the assumed regularity for the
solution ensures that both uand f(x;t;u;u
x
)belong toL 1
lo
(IR(0;+1)):
Our result is the
Theorem 4.1 Under assumptions (H3), Equation (21) has at most one solution in
C 1
(IR(0;+1))\C(IR[0;+1)) for eah initial datum u
0
2C(IR ):
Proof of Theorem 4.1. We suppose, by ontradition, that we have two solutions
u;v 2C 1
(IR(0;+1))\C(IR[0;+1)) of Equation (21). Forevery " >0; we dene
~ u
"
(x;t)= Z
x
0
u(y;t+")dy+ Z
t+"
"
f(0;;u(0;);u
x
(0;))d
and v~
"
inthe same way replaing u by v in the right-hand side of this equality.
By standard arguments in the theory of distributions, one proves easily that u~
"
and
~ v
"
are lassialsolutions of
!
t
f(x;t+";!
x
;!
xx
)=0 inIR(0;+1): (22)
The funtions f(;+";;) satisfyassumptions (H1) and (H2)uniformlyin "; and
(~u
"
~ v
"
)(x;0)= Z
x
0
(u(y;") v(y;"))dy
0
by applying Proposition 2.1, we dedue that u~
"
~ v
"
onverges to 0 loally uniformly in
IR[0;+1).
It follows, onone hand, that the integral
Z
t
0
[f(0;;u(0;);u
x
(0;)) f(0;;v(0;);v
x
(0;))℄d
is well-dened (notie that it was not the ase a priori for R
t
0
f(0;;u(0;);u
x
(0;))d
and R
t
0
f(0;;v(0;);v
x
(0;))d) and letting " go to 0, we obtain that, for all (x;t) 2
IR[0;+1);
Z
x
0
(u v)(y;t)dy+ Z
t
0
[f(0;;u(0;);u
x
(0;)) f(0;;v(0;);v
x
(0;))℄d =0:
To prove the result, we havejust to dierentiate this equality with respet to x: 2
Remark 4.1 : It would be very interesting to be able to prove the above result by
assuming only the solutions to be in W 1;1
lo
. The diÆulty to do that would be in the
integration ofequation (21)to getequation (22). Onlyfew resultsexists inthis diretion
and mainlyfor rst-order equations (f. Corrias [10℄).
Akeyappliationoftheaboveresultonernsthemeanurvatureequationforgraphs
inIR :
Theorem 4.2 Themean urvature equation forgraphs(2) has a unique smoothsolution
u2C(IR[0;+1))\C 1
(IR(0;+1)) for every initial datumu
0
2C(IR ):
Proof of Theorem 4.2. The existene part of the theorem omes from the result of
Eker and Huisken [13℄ (see also[6℄). The uniqueness part is an immediate onsequene
of Theorem 4.1by taking f(p)=artan(p): In this ase (21) readsexatly (2)and, sine
the funtion artan isa smooth nondereasing bounded funtion whih liesin W 1;1
(IR );
thus inC 0;
(IR ) for every 2(0;1);(H3)holds and thereforeTheorem 4.1 applies. 2
5 Appliation to geometrial motions in the plane
We onsider, inthis setion,appliationsfor the mean urvature motionof graphs inthe
plane IR 2
. We rstreall briey somebasi fatsabout the level-set approahinthe ase
of the mean urvature motion.
In this framework, the graph of the initial datum u
0
2 C(IR ) of (2) is represented
as the hypersurfae
0
= f(x;y) 2 IR 2
: u
0
(x) = yg in the plane. We also dene
0
=
f(x;y) 2 IR : u
0
(x)< yg and we take any uniformly ontinuous funtion v
0
: IR ! IR
suh that
0
=f(x;y)2IR 2
:v
0
(x;y)=0g and
0
=f(x;y)2IR 2
:v
0
(x;y)>0g: (23)
If X isthe spae of funtionsw:IR 2
(0;+1)!IR whihare uniformlyontinuous
in IR 2
(0;+1) for all T >0; by results of Evans and Spruk [14℄ and Chen, Gigaand
Goto [8℄, there exists aunique visosity solution v 2X of the geometrial equation
8
<
: v
t
1
v 2
x +v
2
y (v
xx v
2
y 2v
xy v
x v
y +v
yy v
2
x
)=0 in IR 2
(0;+1);
v(;;0)=v
0
in IR 2
:
Moreover, if we dene, forevery t0;
t
=f(x;y)2IR 2
:v(x;y;t)=0g and
t
=f(x;y)2IR 2
:v(x;y;t)>0g; (24)
then thesets (
t )
t0
and (
t )
t0
depend onlyon
0
and
0
but notonthe hoie oftheir
representation through v
0 .
The family (
t )
t0
is alled the generalized evolution by mean urvature of the graph
0
. Anatural issueis the onnetionbetween this generalizedevolution and the lassial
motion by mean urvature. We reall that in general
t
is just dened as the 0-level set
of a ontinuous funtionand therefore it may be nonsmooth and even fatten.
In our ontext, we have
Theorem 5.1 If u
0
2 C(IR ); then, for every t 0; the set
t
is a entire smooth graph,
namely
t
=f(x;y)2IR 2
:y=u(x;t)g;
where u is the unique smooth solution of (2) with initial datum u
0
: Moreover, the evolu-
tion of
t
agrees with the lassial motion by mean urvature in the sense of dierential
geometry.
Proof of Theorem 5.1. From [6℄, we know that, if we start with an hypersurfae
0
whih is anentire ontinuous graph in IRIR ; then,for every t0;
t
=f(x;y)2IR 2
:u (x;t)yu +
(x;t)g;
whereu and u +
are respetively the minimaland the maximal (possibly disontinuous)
visositysolutionof(2). Inthespeialaseofthemeanurvatureequation,weprovedthat
theboundaryofthefront
t
issmooth. Itfollowsthatu andu +
aresmooth. Uniqueness
for(2)inthelassofsmoothfuntions(seeTheorem4.2)impliesthatu =u +
=uwhere
u is the unique solution to (2). Finally,
t
= Graph(u(;t)) is a smooth submanifold of
IR 2
(inpartiular,
t
neverfattens). Inthis ase,the generalizedevolutionoinides with
agreementwith an alternative generalizedmotion). 2
Using the previous geometrial approah, we an state a renement of Theorem 4.2,
namely a omparisonresult whihholds for any visosity sub- and supersolutions of (2).
Corollary 5.1 If u
1
(resp. u
2
) isan upper-semiontinuousvisosity subsolution (respe-
tively lower-semiontinuous supersolution) of (2) and ifu
1
(x;0)u
0
(x)u
2
(x;0)in IR ,
then u
1 u
2
in IR[0;+1):
Proof of Corollary 5.1. This result is an immediate onsequene of a more general
resultintheaseofthemeanurvatureequation: uniquenessforsmoothsolutionsimplies
omparison in the lass of disontinuous visosity solutions (see [6℄ for a proof). To be
self-ontained,we provide ashort proof whihemphasizes the main ideas of the proof.
By Theorem 6.1 in [6℄, sine u
1
and u
2
are respetively an us visosity subsolution
and a lssupersolution of (2), we have
Graph(u
1
(;t))f(x;y)2IR N+1
:v(x;y;t)0g;
and
Graph(u
2
(;t)) f(x;y)2IR N+1
:v(x;y;t)0g :
Essentially, this omes from the preservation of inlusion for sets moving by their mean
urvature inthe levelset approah(see Evans and Spruk [14℄).
Wereallthatv isaninreasingfuntionofybut theaboveinlusionsdonot giveany
information about the relative position of Graph(u
1
(;t)) and Graph(u
2
(;t)) when the
fronts
t (u
0
)develop interior. But,thankstoTheorem5.1,weknowthat
t (u
0
)isexatly
the graph of u(;t), where u is the unique smooth solution of (2) with initialdatum u
0 .
The front does not fatten and is the boundary of
t (u
0
). It follows that
u
1
(;t)u(;t)u
2
(;t) inIR N
;
whih ends the proof. 2
Weonludethissetionwithanextensionofthepreviousresultstosomemoregeneral
equations assoiated to more generalgeometri motions. We onsider
8
<
: u
t
(f(u
x ))
x
=0 inIR(0;+1);
u(;0)=u
0
in IR ;
(25)
where f 2C 1
(IR ):
Theorem 5.2 Let u
0 2W
lo
(IR ): Suppose that
f 2C 1
(IR ) and 0<f 0
(p) C
1+p 2
for every p2IR : (26)
Then(25) admitsauniquesolution u2C 2
(IR(0;+1))\C(IR[0;+1)):Thegeneral-
ized evolution of
0
=Graph(u
0 ) is
t
=Graph(u(;t)) for t 0: It evolves with normal
veloity equal to
V((x;y);t)=f 0
( otan ())
sin 2
()
; (27)
where and denote respetively the angle between the y-axis and the normal outward
vetor and the urvature to
t
at the point (x;y):
Proof of Theorem 5.2. We onsider as above an uniformly ontinuous funtion v
0 :
IR 2
! IR suh that fv
0
= 0g =
0
and fv
0
> 0g =
0
:= fy > u
0
(x)g: Following [6,
Setion4℄, if (26) holds, then
8
<
: v
t f
0
v
x
v
y
(v
xx 2v
xy
v
x
v
y
+v
yy
v
x
v
y
2
)=0 in IR 2
(0;+1);
v(;;0)=v
0
in IR 2
(28)
admitsa uniquesolution v 2X:The levelset approahapplies forthe generalizedevolu-
tion (
t )
t0 of
0
and evolves formallywith normal veloity given by (27).
On a otherhand, usingChou and Kwong [9℄, welearn from[6℄that, againbeauseof
(26), the boundary of
t
is made of the graphs of two smooth solutions of (25), namely
u +
and u : Notiing that (26) implies (H3), we obtain that (25) has a unique smooth
solution; thus u +
= u := u and
t
= Graph(u(;t)): Finally, note that, sine
t is
smooth, (27) holds ina lassial sense. 2
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