Viscosity Solutions of Cauchy Problems for Hamilton-Jacobi Equations

MEMOtES OF SHeNAN

INSTrTUTE OE TECHNObO(;Y

VoL 28,Ne,1,1994

Viscosity

Solutions

of

Cauchy

Problems

Hamilton-Jacobi

Equations

for

Kazuo

KoBAyAsi*

The viscosity solutions of the Cauchy problem u,+H(x,u,Du}=O, u(x,O)=uo(x) inRN, where H: RN ×

RXRN-R isa continuous function, are considered. We _prove an existence and uniqueness theorem

under a condition which ismore _general than the usual one with respect to the u dependence of the

HamiltonianH(x,u,p).Thisgeneralized condition would not necessarily _guarantee that the stationary

problem u+H<x,u,Du}=f in

R"

has a continuous viscosity solution. Our main method isbasedon the

.

technique from nonlinear semigroup theory.

1.

Intreduction

In thispaper we are concerned with the

ex-istence

and uniqueness of viscosity solutions of

the

Cauchy

problem

for

Hamilton-Jacobi

equa-tions

ut+H<x,u,Du)=O in R" ×

(e,T),

(1.1)

u(x,O);uo(x) in R",

(1,2)

in

which

H:

RNxRxRN.R

is

continuous and

Du

denotes

the spatial _gradient

(ux,,

''',ux,Jof

u. Crandall and Lions

[3,4],

Crandall, Evans

and Lions

[1],

Ishii

[5,

6,7]and Souganidis

[lO]

have treated the _problem

_(1.IHI.2)

as well as

the related stationary _problem

u+AH(x,u,Du)=f in

R",

(1.3)

in

which

a

>O andf:

RN-R

is

agiven function. The conditions on the Hamiltonian

H

assumed

in these _papers cited above _guarantee that

there exist both continuous viscosity solutions

of

(1,1)-(1,2)

and

(1.3).

The main _purpose of this _paper isto

general-ize

the usual condition on the u

dependence

of

the Harniltonian H{x,u,P)and to_prove that the

Cauchy

_problem

(1.IHI.2)

has

a unique and

continuous viscosity solution under the

condi-tion

_(see

the condition

(Hl)

below);

however,

the uniqueness of viscosity solutions of the *

_tyasxeE

_tzfiN

SPat ₅

Hi

_1O

n

₂₀

_H"N

stationary problem

(1.3)

is

not necessarily

proved under the condition,

Another purpose of this_paper

is

to _give a

nonlinear semigroup approach to the Cauchy

problem

(1.1)-(1.2}

which

is

not a direct

appli-cation of

Crandall-LiggetVs

theorem, since the

operator

A

in

BUC(R")

defined

formally by

Au=H(

･,u,Du)

is

"not necessarily accretive"

in

BUC(R")

under the condition

(Hl).

Our

main

tools are refinements of the technique

from

nonlinear semigroup theory

{see

[8],

[9]

and

[11]>.

2.

Statements

of Results

We will use the following notational

conven-tions throughout. We will set 9=RN and

de-note by BUC(9) the Banach space of bounded

uniformly continuous

functions

defined

on

O

with themaximum norm

11u11

==sup.Eo

lu(x)1.

We

denote

by

r

the set of

functions

m:[O,oo)-.

[O,oo)

which are continuous, nondecreasing and subadditive and satisfy m(O>=O.

We will not explain the definitionsof

viscos-ity

subsolutions, supersolutions and

solu-tions of

(1.1)

and

(1.3>;

we refer the reader to

Crandall,

Evans

and

Lions

[1]

and

Ishii

[7]

for

the _precisedefinitions.

For

the

Hamiltonian

HEC(9XR

×

9)

consid-ered here we will assume the following

condi-tlons:

(Hl)

There isrEI" such that

S6drir(r)-oo

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mamzNJ(#eee

ag

2sg

ag

1e

H<x,r,p)-H(x,s,P)2-r(r-s)

forall x,pE9 and r,sER with r)s.

(H2)

There isaEIr such

that

)H(x,r.p}-H(x,r,q)1Sa(jp-q1)

for

all x,

_P,qE9

and rER.

(H3)

There

is

mEr such that

{2.1)

(2.2)

IH(x,r,p)-H<y,r,p)ISm(lx-yl(to1+1)}

forallx,y,PE9 and rER.

(H4)

sup.EolH(x,O,O)I<oo.

Our result isas follows:

THEoREM. Let

(Hl>-(H4)

be satisked and

T>O.

11f

uoEBUC(9) then thereexists a unique

viscosity solution uEIC([O,

T];BUC(9))

of

(1.1)-(1.2>.

Remark.

In

[4]

(also

see

[3]

and

[10])

the

above theorem

is

_proved under the conditions

(Hl)',

(H2}',

{H3)

and

{H4),

where

<Hl)'

There isa real number a such that

(2.1)

holds

with

Xr)

==any

<H2)'

For each

R>O

there

is

a _qi?Er such

that

{2.2)

with o=cb?

holds

for

all rER and all x,

p,qE9

with

IPI,lqlSR.

Clearly,

(Hl)'

implies

{Hl)

and

<H2)

implies

(H2)'.

Therefore,

our result _generalizesthe

con-dition

on the u

dependence

of the

Hamiltonian

H

at the expence of strengthening the

condi-tion

of

the

Hamiltonian

H

on

the

gradient of u.

The

condition

{H2)

is,

however,

only needed

in

the proof of

Proposition

3.1

below;

we may

replace the condition

(H2)

by

the usual

condi-tion

_<H2)'

inthe other _parts of

Section

3 below.

On

the other

hand,

it

would seem that the

finitenessof the right derivative of _rat O

in

the

condition

(Hl)'

_plays an

important

role

in

the

arguments used

in

[4],

[3]

and

[10].

We

can,

however, choose a function _rEr which

satis-fies

S8drtr(r)=oo

and r'(+O)=oo

in

the

condi-tion

_(H1),

for

example, _r(r}=-rlogr for O<rS

(2e)J'

and r(r)=rlog

2+{2e)Li

for

r>{2e)-'.

3.

Proof

of

Theorem

We start with the

following

comparison

result

for

(1.3)

which will play an

important

role

in

our arguments.

PRoposlTioN

3.1.

Let

(Hl)-(H3)

be satished,

Let Z>O and u,v,f;gEL"e<9).

Uu

_{and v} are a

viscosity subsolution

of

u+aH(x,u,Du)=f on

9

and a m'scosity suPeTsolution

of

v+ZH<x, v,Dv)=

g on

O,

respectively, _{then we}

have

sup(u*-v*)'SAr{sup(u*-v*>')+sup(f*-g*)'.

0 O 9

Here

r'

denotes

max{r,O} and u*

is

the tipper semicontinuous relaxation

of

u definedby

u'(x)=Iimsup{u(y);ly-xl<e}

forxE9

c.+O

and u*

is

the

lower

semicontinueus retaxation

of

u

dajned

by

u*=-(-u)*.

PRooF.

This proposition isproved in

[5]

under the conditions that u,v,LgEUC(9) and

(Hl)',

(H2)'

and

(H3)

are satisfied. We can not.

however, find the

literature.

which gives the

proof

in

thecase thatu.v,L _gare discontinuous

and

(Hl)'

isreplaced

by

(Hl).

Hence

we will

give a

brief

_proof.

We

follow

the idea

menti-oned

in

[2].

Setz(x,y)==u'(x}-v*<y) andh(x,y)=

f*(x}-g*(y)

for x,yE9.

Let

_gRECi(R)

satisfy

OSgkS1, _gR(r)lr.1as r.oo and _gR(r)=O forrK

R,

LetB,n,E,6E(O,1]andset

G=

{(x,y);

x,yE9,

lx

-yl <n,

lxl

<le}

with rle>R _{satisfying a <BgR(ris),}where a =

sup..yEo z(x, _y).

We

introduce

the

function

which will correspond to w,

in

the condition

(H4)

in

[2]

w(x,y)=A<x-y>S+B+BgR(lxl)

+Ar(supg')+suph'

G G

with

<x>d=(lx12+62)"2,

_"=min

{et2Am<e),

1},A =

rnax

{af,7",

2Am(E)le} _and

B=Am{E)+Zaes)+E.

Comparing

w with z on G ina manner similar

to

_[2],

we will conclude

zgzv onG.

<3.1)

Next we claim that foreachB>O

lim

supzSsupz(x,x)'

{3.2}

n.+o 6 xEO

Indeed,

a' attains itsrnaximum at

(x,,y,)EC.

Since v* is1.s,c.and

bounded

from

below,

by

Baire's

theorem there exists a sequence

{q.}

in

C(9)

such that

_{epi(y)Sq2(y)S}

･･･gep.(y)/v*(y)

ViscositySotutions

of

CauchpuPtoblems

for

Htimilton:ldcobiEquations

foryE9,

Since we may assume thatx,-xo and

1epn<xn)-epn(y)i-O

as _nyL.+O,one has

lim

SUPZ'K

liM{U*(Xn)-epn(Yn))'

n-+O C n.+O

S

(U*(xo)

-g.

(x,))4

.

By

letting

n--oo we

have

(3,2).

It

is

clear that

(3,2)

with z replaced by h

is

also valid.

We

let

x[=y

in

(3.1)

and then 6-.+O, E.+O,

ny-+O,

B･+O

and

R.-oo

in

thisorder. Then,

using

{3.2)

we have the desired

inequality.

Q.ED

Let

RE(O,112r(1)>,

T>O and IVh'={1,2,･･･,

[TXA]+

1},

where

[TIA]

denotes thelargest

inte-ger part of T/A. Let uoEL"'(9) and u2EiLO'(9),

kEIVIU

{O},

be

the functions such thatuao=uo and u2 isa viscosity solution of

u2+aH(x,uk,Du2)=uZ.i on'9.

(3.3)

Indeed,by

<H1)

and

(H4)

the equation

(3,3)

pos-sesses viscosity sub- and supersolutions which

take constant values on

9

whenever u2L]E

LOO<9). Thus, there exists a viscosity solution u2 of

(3.3)

inLO'(9)by virtue of

[7,

Theorem

3.1].

Now

consider the

Cauchy

_problem of ODE

ep'<t}:-r(q<t)),t20, ep(O)-a>O.

Sin

¢e r(r)

is

continuous and grows at most

linearlyas r.co, Peano's theorem assures the

existence of the_global maximal solution

_ip{t;a),

tE[O, oo). We set

tu(a)= ¢

(T;a).

(3.4}

Then

we have that_¢

<t;

a)Sto(a)

for

tE[O,

T]

and

lim..+otu(a)=O since

S6dr/r(r)=

oo.

LEMMA

3.2.

LetuoEBUC(9).

71here

exists a

constant

M>O

such thatOS(ui)Kto(MA)

for

leE

IVh,where

OS(u)=sup.Eg(u*(x)-u*(x)).

PRooF. By Proposition

3.1

we

have

oS(tt2)sAr<OS<uAk)>+OS(u2-i)

(3.5}

which _yields that

OS(u2>K(1-Zr(1))-'{Zr(1)+

OS(u2-i)}-<e2ti"T7"r(1)since

OS(uo)=O.

Define

a

step

function

dii

on

[O,

T]

by

¢i(O)=O and ¢i(t)=

OS(u2)

for

tE((k-1)1,hA],

kEIVI.

Then,

summ-ing

(3.5)

yields

that

ipi(t)SgS:r(ipi(T))dT+AM,

tE[O,T],

where M=r(e2ri')TTr(1)).Therefore,by the

max-imum

principle one

has

that

ip,(t)gip(t;ZM)S

to(MZJ, tE[e, T].

Q.E.D.

For A,tt>O we set

ak,f=max{sup(u2*-uf*)',sup(u,"･*-u2*)'}

9 O

and definestep

functions

uA and u"'A on

[O,

T]

by

u"(o)=u"･"(O)=o and uA(t)=u2 and u"'a(t>=:u"(hR)

fortE((le- 1)Z.hA],

kENI.

LEMMA 3,3.

Let

ueEBUC(9) and zuEC,03(9}.

7]Pien

thereexist constants

C

and

C.

and _p,p.EI"

such that

C

and _p are indePendent

of

tv but

C.

and p.,may

depend

on _w and that

ak,i<LC11uo"w11+C.A,i kA

+S, _{r(11u"(T}-di'A{dll)dT} +ittlo(11uo-wll)+p.(A+lt) +6-'p.<TYL,j+p.<26)}

(3.6)

foratl6E(O,T12),A,uEE(O,6),leEAII,1'EIVI,.

flere

Aj=

{(lez

-ju

)2

+

hz2

+ict2}u2.

PRooF.

In

what

follows

C,

C.

andp, _p. tvill

denote

various constants and _.functions in

L

re-spectively, such that

C

and p are

independent

of

w

but

C.

and _p. may depend on w. It

follows

from Proposition

3.1

that supo(uk*-w>'s

(1

-Ar(1))Li

{Zr(1)+supo(u2Ei-f)'}

where

f=w+

AH<x,w,Dw).

Hence we have that

sup(u2*-tv)'SCIIue-wlL+C.kA.

(3i7)

o

Similarly,supo(w-u2*)' has the same bound.

Thus one

has

that

11u211SC.

for kEIVI, By

Proposition 3.1,foreach iEIVI

dh=max{sup(u2fi-u2*)',sup<uZ*-(u:+i)*)'}

m o satisfies k

dkSdk.i+Ar<dk)gdo+RZr(dj),

j--1

which together with

(3.7)

and Lemma 3.2

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mamrrvft\$eet

eg

2sg

rg

1e

k-1

11u2+i-uillSAi+Zr(11u,A･+i-uSl)),

J'--1

where

Ai=Clluo-tvH+C.iA+p.<A).

Therefore,

by the maximum _principleone has

11u2+i-u211gw(zLi)

for

le,le+iEIVA.

{3.8)

Next,

from

Proposition

3.1

one has that

(JL+Lt)ahj-<Aak,j-i+Ltak-i.j+Altr(ak,P･

On

the other

hand,

by

(3.8)

and

Lemma

3.2

one '

has that

r(ak,j)Sr<11uX(leJL)-u"'A(leA)11)+A,. whereA2=p(11uo-wll)+p.(A+")+p.(1leA-lul). Thus we obtain

(1+U>ak,j-<Zak,j-1+ptak-1,j

+ztt

{r(liuA(kz}-u#･a(lez)ll)+A2}

.

(3.9)

To

prove

{3.6)

we

first

note that

(3.6)

is

valid

for le=O or _1'=O by

(3.7).

Therefore, by using

{3.9)

and

the

usual induction arguments as in

[8,

9]

we conclude that

(3.6)

is

valid

for

all

hEIVI

andjEIVI.

Q.E,D.

PRooF oF THEoREM.

We

first

fix

A,st

>O

arbi-trarily,We then take

6=(2+">it4

and

integers

k,1'such thatjuE((h- 1)A,kA]in

<3.6),

Then,

by

using

(3.8)

one easily obtains

11uA(t)-tt"'"{t)HKp(11uo-tvll)+p.(Z+lt)

t

+S,r(IIuA<T)-u"'A<r>11nT

with suitablep,p.El-: Note that CoeO(9)isdense

in

BUC(9), and _{w can} be taken arbitrarily from

Cff(9).

Hence

it

follows

from

the fact that

So'

dnyr(r)

= _oo

that

lim

sup

11uA(t)-u"'A(t)11=O,

A,#-+O tE[O,T]

which together with

(3.8)

implies

that

there exists a

function

uEC([O,T];Lee(9))

(see

[9,

1O])

such that

lim sup

llu(･,t}-uA{t)11=O.

A-+O tE[O,T]

Since OS(u<･,t})=O

for

tE[O,T]

by

Lemma

3.2,

u must lieinBC(9 ×

[O,T]),

which

denotes

the set of

bounded

continuous functions defined on

9

×

[O,T].

Then,

the _proof of

[1,

Proposition

5.2]iseasily adapted to_prove thatu isa

viscos-itysolution of

(1.1)-{1.2).

Therefore the _proof

of the theorem will

be

completed

if

we _prove

the

following

lemma,

Q.E:D.

LEMMA3.4.

Let(H1)-(H3)besatished.

UuoE

BC(9)

then

(1.1)-(1.2)

has

at most one viscosity

solution u

in

BC(9

×

[O,T]).

Moreoven

of

uoE

BUC(9)

then uEC([O,

T];BUC(9)).

OuTmNE

oF

PRooF.

Let

u and v

be

two

vis-cosity solutions of

(1.1)

in

BC(9

×

[O,

T])and _g(x,

y,t)=tt(x,t)-v(y,t).

Let

gR,G

and so on

be

the

functions as

in

the proof of

Proposition

3.1.

Consider

the

function

z(x.y. t>=

{A

<x

-y>"6+B+BgR(lxl)}

(t+

1)

t

+S,r(S.Upz(',`,s)')ds+s.upz(･,-,O)',

where _pt:=:_min

{et2(T+

1)m(E),

1}

,

A

= max

{afn",

2m(E)tE}

andB=m(E)+aue(T+1))+E.

By

the

comparison argurnent as

in

[2]

we obtain

zgzv onGx(O,T].

(3,10)

In

(3.10)

we letx=y and then 6-.+O, e.+O,

n.+O,

B.+O

and

R.oo

in

thisorder.

Then

we have thaton

9

×

[O,

T]

z(x,y,t>'K?g.pz(x,x,O)'+S:r(?g.pz(x,x,s)')ds.

Here we

have

used

(3.2)

with z(x,y> replaced

by

z{x,y,t),

This

implies

that ifz(x,x,O)'=O then

z(x,x,t)'=O

for

tE[O,T].

Hence

we

have

the uniqueness of solutions.

Finally,

we

let

u=v in

(3.1O)

and

then

let

Tle-.

oo, _n--+O,

6-+O,

_e.+O,

B-+O

_and

R-oo

in

thisorder.

Then

one obtains

that

the

fvnction

g(t)=limn-+osupi.-yi<nlu(x,t)-v(y,t)1satisfies

g(t)Kg(o)+S:r(g(s))ds.

Since

g(O)=::O

by

uoEBUC(9), we

have

that

g<t)

=O _and hence _{u<･,t)EBUC(9)}

for

tE[O,

T].

Q.ED.

References

[1]

M. G. Crandall, L C.Evance and P.L.Lions,

Some Properties

of

viscosas solutions

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tonilacobiequations, Trans.Amer. Math. Soc.

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forHlimitton-jocobi

Equations

282

(1984),

487-502.

[2]

M.G.Crandall,H,Ishiiand P.L.Lions,

ness

of

viscosity sotutions

of

Hamiltonilk]cobi

equations revisited,

J.

Math. Soc.

Japan

39

(1987),581-596.

[3]

M. G.Crandalland P.L.Lions, ViscosiC),

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Hdmiltonzlirtcobiequations, Trans.

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[4]

M.G.Crandalland P.L.Lions,On existence and

uniqueness

of

sotutions

offidmilton-facobi

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[5]

H. Ishii,Remarks on the existence

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viscosity

sotutions

of

Hlrimiltonzlacobiequations, Bull.

Fac. ScL Engng. Chuo Univ.26

(1983),

5-24.

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H.Ishii,Uitiqueness

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unbounded solzations

of

Hamiltonzlacobiequations, Indiana Univ. Math.

[7]

[8]

[9]

[le]

[11]

J.33 (1984),721-748.

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Hamilton:lbcobi

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(1987L

369-384.

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Non-linearevotution oPerators in Banach sPaces,

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J.

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345-390.

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Tanaka and K.Kobayasi,Generationtheorem

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