A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

H ^ITOSHI I ^SHII

A boundary value problem of the Dirichlet type for Hamilton-Jacobi equations

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 16, n

^o

1 (1989), p. 105-135

<http://www.numdam.org/item?id=ASNSP_1989_4_16_1_105_0>

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for Hamilton - Jacobi Equations

HITOSHI ISHII

0. - Introduction

In this paper ^we consider the

boundary

^value

problem

^{for Hamilton -}

Jacobi

equations:

in

or on ~ ⁱ

Here 11 is an

open subset

^{and h : an --~ R}

are

given functions, u : fi

^{--~ R is the} unknown and Du denotes the

gradient

^of

u.

The

meaning

^of

(solution of) problem (HJ) - (BC)

will be made

precise

ⁱⁿ

Section 1

by modifying

^{the notion of}

viscosity solution,

introduced

originally by

M. G. Crandall - P.L. Lions

[7],

^{so that the}

boundary

condition is

appropriately

taken into account. There are

already

^several attempts of similar modifications of the notion of

viscosity

^{solution. For this we refer to} R. Jensen

[15],

^P.L.

Lions

[18],

M.G. Crandall - R. Newcomb

[8],

H.M. Soner

[20],

^P.E.

Souganidis [21],

^1.

Capuzzo

Dolcetta - P.L. Lions

[4]

^and G. Barles - B. Perthame

[1].

^Our

introducing

^the

boundary

^condition

(BC)

^is

strongly

^motivated

by

^the

analogy

to the Neumann condition for Hamilton - Jacobi

equations

^{in P.L. Lions}

[18].

Problem

(HJ) - (BC)

^is

important

ⁱⁿ deterministic

optimal

^control

theory.

The value functions of exit time

problems satisfy (HJ) - (BC),

^{where the first}

order PDE in

(HJ)

^is the so-called Bellman

equation,

^{in the}

viscosity

^{sense, and}

they

^are characterized to be

viscosity

^{solutions of}

(HJ) - (BC),

^under

appropriate hypotheses.

^We prove ^{these in} Sections 3 and 5

where, denoting H(x,

^u,

p) - u again by H (x, u, p),

^{we treat the}

^problem:

in

Pervenuto alla Redazione il 25 Novembre 1986 e in forma definitiva il 7 Febbraio 1989.

and

or on

The

boundary problem

^for

^(HJ),

with the Dirichlet condition u ⁼ h, ^{is not} solvable in

general.

^A

compatibility

^{condition on h is}

required

^to be satisfied in order that the Dirichlet

problem

^for

(HJ)

^{is solvable}

(see

^{P.L. Lions}

[16]),

^and

the condition on h is

usually

^{hard to} check. See also the recent work

[9] by

^H.

Engler.

^{On the other}

hand, problem (HJ) - (BC)

^is

solvable,

^{in a weak} sense, under

quite general hypotheses ^(see Proposition

^{1.3 in Section}

1).

Moreover the

uniqueness

^and existence of a continuous

viscosity

^{solution of}

(HJ) - (BC)

^is

established under suitable

assumptions

ⁱⁿ Sections 2 and 4. Therefore

problem (HJ) - ^(BC)

^{seems, at} least for the

author,

^{a natural}

replacement

of the Dirichlet

problem

^for

^(HJ).

^Problem

(HJ) - (BC)

also arises

naturally

ⁱⁿ connection with

elliptic singular perturbation problems

with the Dirichlet

boundary

^{condition or}

with the

vanishing viscosity

method. In this direction see

Proposition

^{1.2 and}

refer to H. Ishii - S. Koike

[14]

^{for an}

application

^{of our results to}

elliptic singular perturbation problems. Finally

^we remark that G. Barles - B. Perthame

[2]

^and P.L. Lions

[19]

^have

recently

^treated

(HJ) - (BC) independently.

1. -

Viscosity

^solutions

In this section we define the

viscosity

^{solution of the}

boundary problem

for a second order PDE:

in and

or on

and then present ^{some of its}

properties.

^{Here n is an open} subset of is a subset of the

boundary

^{F : n u E x R x x - R, where}

^~N

denotes the space of real N ^{x N}

matrices,

^{and are}

given functions,

^{u : n u E -~ R is the}

unknown,

^and

D2 u

denotes the Hessian matrix of u. When F is

independent

of the last

variables,

^i.e.

F (x,

^{r, p,}

~)

⁼

F (x,

^r,

p),

and B ⁼

F, problem (1.1) - (1.2)

^{will be}

simply

^{written as}

on

Our main interest here is to

study problem (HJ) - (BC). However,

^{it seems} natural to see the connection between

viscosity

solutions of

(HJ) - (BC)

^and

the

vanishing viscosity

^{method in}

light

^{of the}

stability

property ^of

viscosity

solutions of

(1.1) - (1.2).

Let

S,

T and U be subsets of

satisfying

^{S c U and S c T c}

^S’.

^For

a function

f :

^{U --; R u}

{ - oo, oo ~,

^{we define}

by

sup

This function

(liS, T)*

^{is upper} semicontinuous

(u.s.c.

ⁱⁿ

short)

^on T, and we call it the u.s.c.

envelope of f

restricted to S on T. We note /1 that

(flS,T)*(x) ~! f (x)

^{for x E S and}

^that,

^if

^f

^is u.s.c. on S, ^then

I

( f ~S, T)# (x)

⁼ for x E S. We remark

that,

^if

( f I S, T)* (x)

^{oo for}

x E T, ^then

We define the lower semicontinuous

(l.s.c.

ⁱⁿ

short) envelope

of f

restricted to S on T

by (fI8, T ) * = - (- flS, T ) * .

^{When S = T =}

^U,

^we

write

f *

^and

f * , respectively,

^for

(fI8, T ) *

^and

(f I S, T ) * .

We call a function u : 11 u E -~ R Li a

viscosity

subsolution of

problem ( 1.1 ) - (1.2)

^if

u* ( x )

^{oo, for x E fl u E, and whenever p E}

and u* - _p attains its local maximum at a

point y En

^u

E,

^then

and

or if

Similarly,

^{we call a function a}

viscosity supersolution

of

( 1.1 ) - (1.2)

^if

u* ( ~ ~

&#x3E; - oo, for x and

whenever p

^E

and _{u* - p} attains its local minimun at a

point

y E 11 u E, ^then

’

/

if and

A

viscosity

solution of

(1.1 ) - (1.2)

is defined to be a function on 11 u E which is both a

viscosity

^{sub- and}

supersolution

^of

( 1.1 ) - (1.2).

^{Note that}

this definition makes sense even for functions

F, B

which take values in R u

(-cxJ, cxJ),

and that if

F(x, r, p, ç)

^is

independent

^of

~,

then the space

of "test

functions",

^{can be}

replaced by

^the space

We say also that u satisfies 0 in n and

B ( x, u, Du)

^{0 or}

F ( x,

^u,

^Du, D2 u)

^{0 on E}

(resp., F ( x,

u,

Du,

^{0 in H and}

B ( x, u, Du) &#x3E;

⁰

or 0 on

E),

^{in the}

viscosity

sense, if u is a

viscosity

subsolution

(resp., supersolution)

^of

(1.1) - (1.2).

Of _course, this definition is a modification of the

original

^{one introduced}

by

M.G. Crandall - P.L. Lions

[7]

^and

by

^{P.L. Lions}

[17]

^{to the}

boundary

^value

problem.

^The

boundary

^condition

(1.2)

^{is motivated}

by

^{the work}

[18] by

^P.L.

Lions

concerning

the Neumann

problem

for Hamilton - Jacobi

equations.

^{In our}

notation,

the Neumann condition for

(HJ),

ⁱⁿ

[18],

^can be written as

or on

where v denotes the outward unit normal of an

(assuming

^it

exists).

^We

should also remark that G. Barles - B. Perthame

[2]

^{and P.L. Lions}

[19]

^have

independently

introduced the same notion of

viscosity

solutions for

(HJ) - (BC)

as ours. We refer to H.M. Soner

[20],

^I.

Capuzzo

Dolcetta - P.L. Lions

[4]

^for

another type ^{of the}

boundary

^{condition for} Hamilton - Jacobi

equations

^{and to}

M.G. Crandall - H. Ishii - P.L. Lions

[6],

^{H. Ishii}

[10,12,13]

^{and G. Barles -}

B. Perthame

[1]

for the work related to discontinuous

viscosity

^{solutions and}

Hamiltonians.

PROPOSITION 1.1. Let E be an _open subset

of

^{Let S be a}

family of viscosity

subsolutions

of (1.1) - (1.2).

^Set

u(x) =

^sup

{ v ( x) :

^{v E}

S }, for

x e n u E. Assume u is

locally

^bounded

from

^above on fi U E. Then u is a

viscosity

subsolution

of ( 1.1 ) - (1.2).

This is a

generalization

^of M.G. Crandall - L.C. Evans - P.L. Lions

[5, Prop. 1.4]

^{and H. Ishii}

[13, Prop. 2.4].

^The

proof

^{of this}

proposition

is similar

to that of

[13, Prop. 2.4],

^{and we omit}

giving

^{it here. An}

analogous

^assertion

holds for

supersolutions.

^{To see}

^this,

^observe that u is a

viscosity

subsolution of

(1.1) - (1.2)

^{if and}

only

^{if v = -u is a}

viscosity supersolution

^of

(1.1) - (1.2),

^{with F and B}

replaced, respectively, by

^{the functions}

and

PROPOSITION 1.2.

(Stability of viscosity solutions).

^{Let E be an} open subset

of

^{an. For 0 E} 1, ^let

be

given functions.

^Assume

that, for

^{each 0 e 1, Us is a}

viscosity

subsolution

of ( 1.1 ) - (1.2),

^with

F,

^and

B,

ⁱⁿ

place of

^{F and B,}

respectively.

^Set

for

^and

.for

^{. Assume u is}

locally

^bounded

from

^{above on 11 u E.}

Then u is a

viscosity

subsolution

of ( 1.1 ) - (1.2)

with these F and B.

This

generalizes

^{a result of G.} Barles - B. Perthame

[1,

^Theorem

A.2].

Of _course, we have a

proposition

^{similar to the above for}

supersolutions.

^We

can prove this

proposition

^{as in the}

proof

^of

[1,

^Theorem

A.2],

^{and we do not}

give

the details here.

The

following

observation is useful in

applications

^of

Proposition

^{1.2 to}

elliptic perturbation problems

^{or the}

vanishing viscosity

^method.

are continuous and that u E

u £)

^{n satisfies}

F ( x,

^u,

Du, D2 u)

^{0 in}

n and

B(x, u, Du)

^{0 on E, in} the classical sense.

Assume,

ⁱⁿ

addition,

^that

- F is

elliptic,

^i.e.

for

( x,

^{r, p,}

ç) E nxR ^{xMN and p}

^E

C2(n)

^such

that p

^{attains its maximum}

at x, and that B satisfies the condition:

B ( x, r, p) +D~(~c)) &#x3E; B ( x, r, p) ,

^for

(x, r, p)

^{E E x R x E}

u £)

^such

^{that p}

attains its maximum at x. Then u is a

viscosity

subsolution of

( 1.1 ) - (1.2).

^From

Proposition

1.2 we see, for

instance,

^the

following.

^and

h : all ^--~ R be continuous. For each 0 e 1, ^let Us E

C (fi)

ⁿ

satisfy -

^{+ 0 in}

n,

where A denotes the N dimensional

Laplacian,

onafi, in the classical sense. Set ⁼ _{lim sup}

r10

0 e

r},

^{and assume}

u(x)

^{oo, for x}

^{e fi.}

^{Then u is a}

viscosity

subsolution of

(HJ) - (BC).

PROPOSITION 1.3.

(Existence of viscosity solutions).

^{Let E be an} open subset

of

^{an. Let}

f

^and g

be, respectively, viscosity

^{sub- and}

supersolutions of

( 1.1 ) - (1.2).

^Assume

f

^s g ^on 11 U E and that

f

^and g ^are

locally

^{bounded on}

n U E.

by

= sup

viscosity

subsolution

of ( 1.1 ) - (1.2),

^v g ^on n U

E).

Then u is a

viscosity

^solution

of ( 1.1 ) - (1.2).

This extends Theorem 3.1 of H. Ishii

[13].

^{We leave it to} the reader to prove this

proposition

^{as the}

proof

^{is a}

simple

modification of that of

[13,

Theorem

3 .1 ] . "

2. -

Comparison

^of

viscosity

^solutions

. Hereafter we

study problem (HJ) - (BC).

^{We need the}

following assumptions

^{on H and fl.}

(H 1 )

^{For each}

(x, p)

^x

^the

^{function u ~}

H (x, u, p)

^is

nondecreasing

on R.

(H2)

^{There is a} continuous

nondecreasing

^{function rral :}

[0,oo)

--&#x3E;

~0, oo), satisfying

^rral

(0)

^{= 0, such that}

for and

p E JR. N.

(H3)

^{There is a} continuous

nondecreasing

^{function m2 :}

[0, 00)

^-

[0, 00), satisfying m2 (0)

^{= 0, such that}

and

(H4)

^{There is a} continuous

function n : fi -_+

^{and a constant b} &#x3E; 0 such that

for and

(where B (x, 6)

^{is the usual}

sphere

^{centred at x with}

radius 6).

THEOREM 2.1. Assume 11 is bounded and that

(H 1 ) - (H4)

^{hold. Let u}

and v

satisfy, respectively,

in

on

where a &#x3E; 0 is a constant, and

in

on

in the

viscosity

^{sense. Then:}

(i) If

^{u and v}

are continuous

^at

^{points of}

a11 and h is continuous on an, then u v

on fi.

(ii) If

^u

(resp., v)

is continuous at

points of a11,

^{h is l.s.c.}

(resp., ^u.s.c.)

^on

5n and u h

(resp.,

v &#x3E;

h)

^{holds on} an, ^{then u v}

on-a.

REMARK 2.1. As the

proof

^below

^shows, the

^above

theorem

^is still valid if the

inequality

ⁱⁿ

(H3)

^{is satisfied}

only

ⁱⁿ

a

subset

of 0 x

^{R x}

^{R N}

^{of the}

form U x R x where U is a

neighbourhood

^{of a11 in n.}

REMARK 2.2. Of _course, it is

possible

^{to formulate a}

comparison

^theorem

like Theorem 2.1 for unbounded domains.

The

proof

^{below is a} modification of the arguments ^{in the}

proof

^{of H.M.}

Soner

[20,

^Theorem

2.2]

^{to the current}

problem.

PROOF.

Replacing

^{u and v}

^by

^{u* and v*, we may assume that u is}

u.s.c. and v l.s.c. on

fi. _First,

we prove assertion

(i).

^{To this}

end,

^we suppose

max(u - v )

&#x3E; 0 and will obtain a contradiction in each of the

following

^three

n

possible

^{cases. We will use} the notation:

and

CASE 1. Set

r ^E R and _{p, q} E

Then H

^{is l.s.c.}

by (H 1 )

^{and w satisfies} in x

n,

^{in the}

viscosity

^sense.

Since

R=,(~r,p) &#x3E; limH(x,s,p)

_slr ^and

~*(a:,r,p) limJ?(a:,5,p),

_air ^for

by ^{(ill) -} (H3),

^{we have}

’

for _{r, s} e R

satisfying

^r &#x3E; s and

Let

2 ,

^{2 and set}

A=max(u-v)

^{a ’ and}

for Then we have

for and

by (H2). Also,

^{we have}

for I

for and

for witt

Now we fix 6 so small that

max(u - v )

&#x3E; A + 26 and then e so small that

and if

and Ix - yl e ^(this

^is

possible

^{since w is u.s.c. and}

w(x,

^{A, for x E}

an).

Observe that w - p attains a

positive

^{maximum at some}

point (~, y) E ~ (~)

since w is u.s.c. on

A(e)

^and

max

_{0 A (s)}

^{(w - ~p)}

^{0. Thus}

^(2.3)

^and

^{(2.5), together}

with

(2.4), yield

a contradiction.

CASE 2.

max(u - v) = (u - v) (z)

^and

v (z) h (z),

^{for some z E af2. Let}

and be as in

(H4).

^{Define 1}

by

where e is a small

positive

number. Let

(~, y)

^{be a}

point ^{of fi x n}

^{such that}

~~ (x, y) -

^max

^4D,.

^{Let us assume e b. Then z +}

^{by (H4),}

^and

hence

~~ (~, y) &#x3E; ~~ (z -~ ~r~ (z), z) . 5x5

^This

yields

where _m3 is the modulus of

continuity

^{of u and Therefore}

and so

for some constant

C2

&#x3E; 0.

Using again ^(2.6),

^we get

From

this,

^{we see}

and

for some real-valued continuous function _m4 on

~0, oo) satisfying m4(0) =

^0.

Since

by (2.8) and, choosing

^{0 b small}

enough, ~r~ ( y) - ~ (z) ~ 1+ M4 (s) -

^{b, for}

0 e

b 1,

^{in view of}

(2.9),

^{we see from}

(H4)

^{that Y E} fi, ^{for 0 e}

b 1.

Since u satisfies

(2.1),

^{in the}

viscosity

^{sense, we thus find}

for

Using (2.6), (2.7)

^and

(2.9),

^{we find}

Hence, selecting

⁰

b 2

^{b small}

enough,

^{we see that if 0 e}

b 2 and y E

^{Since v solves}

(2.2),

^{in the}

viscosity

^{sense, we now see}

Choose 0

b3

^{min so} small that for

0

b3.

^We henceforth assume 0 e

b3.

^Now

(2.6)

guarantees

u(()

&#x3E;

V(Y),

^{for 0 e}

^{b 3 . Setting 6}

^{= and}

^{combining (2.10),}

(2.11)

^and

(2.4),

^we get

Thus, using (H3), (H2), (2.7), (2.8)

^and

(2.9),

^{we find}

this

yields

^a contradiction

by selecting

^{e small}

enough.

CASE 3.

max(u - v) = (u - v) (z)

^and

u(z)

&#x3E;

h(z),

^{for some z E Set}

_

n

_

h = -h on afi, ^onil,

^and

for

As

^remarked

^before,

^{u and v,}

^solve, respectively, (2.1 )

^and

(2.2)

^{with H}

and h

in

place

^{of H and}

h,

^{in the}

^viscosity

^{sense. We}

^easily

^{see that}

(HI) -

(H3)

^are satisfied for H = H and that

max(v - 5) = (u - v) (z)

^and

v(z) h(z).

n

Thus the present ^case is reduced to Case

2,

^{and we have a} contradiction.

Now we turn to the

proof

of assertion

(ii).

^{If we assume}

max(u - v)

&#x3E; 0 n

and argue ^as in the

proof

of assertion

(i),

^{then the}

assumption u

^{h on}

(resp.,

v &#x3E; h on

an) eliminates,

^{from the} arguments, ^the

possibility

^that

Case 3

(resp.,

^Case

2)

^{occurs. Meanwhile we}

only

^{need the}

following continuity

of _{u, v} and h: the

continuity

^{of u}

(resp., v)

^at

points

^{of an} and the lower

(resp., upper) semicontinuity

^{of h in the} arguments ^{of Cases 1 and 2}

(resp.,

^{Cases 1}

and

3).

D

A more

precise

result is needed in Section 4.

THEOREM 2.2. Let a be a

positive

number and h : R be l.s.c..

Assume fl is

bounded

^{and that}

^{(HI) - (H3)}

^{hold. Let u and v,}

respectively,

^be

u.s.c. and l.s.c. on

fi,

^and

satisfy (2.1)

^and

(2.2),

^{in the}

viscosity

^{sense. Let}

r be an open subset

of

an, ^{and assume:}

(i)

^u v, on

anBr,. ^{(ii) u}

^{h, on}

r; and

(iii) for

^{each z E r, there is a}

positive

^{number b, a bounded} sequence and a _sequence C

(0, 00) converging

^to 0, such that

for

and lim _n-00

u(z

^{+ =}

u(z).

^{Then u v on}

^fi.

The

proof

of this theorem is similar to that of Theorem 3.1. We

just

present its outline here and leave the details to the reader.

OUTLINE OF PROOF.

Suppose max(u - v )

&#x3E; 0. Let z

e n

be a

point

j n

where the maximum of u - v is attained.

By assumption, z

^{e H U F. If}

max(u -

^an

^{v ) max ( u -}

n

^v),

^{then the same}

^argument,

^{as in Case 1 of the}

^proof

of Theorem

2.1, yields

^a contradiction. If z E r, ^{then we} go ^as in Case 2 of the

proof

^{of Theorem}

2.1,

except ^{that we now use}

with

instead of and will get ^a contradiction. Thus ⁱ

COROLLARY 2.1. Let S~ be bounded and conditions

(H 1 ) - (H4)

^be

satisfied.

Let u and v :

0 -+

R be,

respectively, viscosity

subsolution and

supersolution of (HJ)’ - (BC)’.

^{Then the same} assertions, ^as

(i)

^and

(ii) of

^Theorem

2.1,

^hold.

REMARK 2.3. We have an

assertion,

^{similar to this}

collorary,

^which

corresponds

^{to Theorem} 2.2. This remark also

applies

^to

Corollary

^{2.2. below.}

-

PROOF. Let e &#x3E; 0 and define _{u E}

by

^{Then Us}

solves

in

on

in the

viscosity

^{sense. From} Theorem 2.1 we find that v on

0

in all cases.

Sending e 10,

^we conclude the

proof.

^D

COROLLARY

^{2.2. be} bounded and

(HI) - (H4)

^be

satisfied.

^{Let u and}

v : 0 -+

R

be, respectively, viscosity

^{sub- and}

supersolutions of (HJ) - (BC).

Assume moreover that p --&#x3E; H

( x,

^r,

p) is

^{convex on}

^{R N for}

^each

^{( x, r)}

^E

ⁿ

^{x R,}

and that there is a

function 0

^{E cl}

(0)

^{and a constant a} &#x3E; 0

for

^which

for

Then the same assertions, ^as

(i)

^and

(ii)

^{in Theorem} 2.1, ^hold.

PROOF. Note that h is bounded below on afi, ^{in all cases. Therefore we} may

h(x),

^{for x E}

by adding

^a

negative

constant to h, ^if necessary.

Then 0

^{is a}

viscosity

subsolution of

(HJ) - (BC)

^{and hence the}

maximum of u

and 0

^taken

pointwise,

^{is a}

viscosity

subsolution of

(HJ) - (BC) by Proposition

1.1. Now let 0 6 1 and set ⁼

8 u V ~ ( x ~ -+- ( 1- 8 ) ~ ( ~ ~ ,

for x E

n. By

^the

convexity

^{of H, we see that} wo satisfies

in

on

in the

viscosity _sense.

^{To check}

^this, let p

^E and suppose attains its maximum

at yEn. Suppose

^further

0

(this implies if y E

^{otherwise we are done. Then}

attains its maximum at y, and hence

Thus

by (2.12);

this proves

(2.13).

^Now

Theorem 2.1

guarantees ^{v, on}

fi.

Sending

⁰

T

1, ^{we see that} v &#x3E; u

V 0

&#x3E; u on

fi.

D REMARK 2.4.

If

^u

(resp., v)

^is

Lipschitz

continuous ^on

fi,

in assertion

(ii) of

^Theorem

2.1,

^{then we have the same} conclusion with the weaker

assumption (2.14)

^{below in}

place of (H2)

^and

(H3).

(2.14)

^{For each R} &#x3E; 0, there is a continuous

function

mR :

[0, 00)

^-

[0, 00), satisfying

mR

(0)

^{= 0, such that}

for and ;

Analogous

^{remarks are valid for} Corollaries 2.1 and 2.2.

PROOF OF REMARK 2.4. Assume all the

hypotheses

^{of Theorem}

2.1,

_except

(H2) - (H3). Assume

^further

^{(2.14) and,}

^for

definiteness,

^{that u is}

Lipschitz

continuous on

n,

^with

Lipschitz

^{constant C} &#x3E; 0. Set R ⁼ C + a, and define

Set

for E n x Then G satisfies

(H2)

^and

(H3),

^with

appropriate

functions _mi and _m2.

Also,

^{u and v}

satisfy, respectively, (2.1)

^and

(2.2),

^with

G in

place

^of H, ^{in the}

viscosity

^sense. Theorem 2.1 now guarantees ^our

assertions. D

3. - Value functions of exit time

problems

In this section we will show that value functions of exit time

problems,

in deterministic

optimal

^control

theory, satisfy

the associated Hamilton - Jacobi

(Bellman) equations.

Let 0

be,

^as

usual,

^an open subset A a compact

topological

space,

f

^a real-valued function x

A, g

^{a function x A}

into RN and h

a real-valued function on an. We ^assume:

(A 1 )

^Functions

f, g

^{and h are} continuous. Moreover, ^{there is a} constant L &#x3E; 0 such that

for and

Let ,~ denote the set of

controls,

^i.e.

For X E and a

E A,

^we consider the initial value

problem

The

unique

solution of

(3.1 )_will

^{be denoted}

^by

⁼

^{X ~t;}

^x,

^a).

^{The exit}

times T from n and 7 from 11 are

defined, respectively, by

and

Associated with these are cost functionals:

and

With these cost functionals at

hand,

^the main purpose of

optimal

^{control is}

stated

^as follows: Find controls or sequences of controls which minimize

a)

and

J ( x, a )

^{over A.} Our interest

here, however,

is restricted to

characterizing

value functions:

and on

~

as

viscosity

solutions of the

boundary

^value

problem

in

on

where

The

first order PDE in

(3.3)

^is called the Bellman

equation.

^{Note that}

H : 0

x I1~ satisfies

(H 1 ) - (H3),

^under

assumption ^{(Al ).}

LEMMA

3.1. (Dynamic programming principle).

^Assume

(A 1 ).

For any t &#x3E; 0 one has

and

where,

for

any subset B

of A,

1B denotes the characteristic

function of

^B.

We refer to P.L. Lions

[16]

^{for a}

proof

of this lemma.

THEOREM 3.1. Assume

(A 1 ).

^The

functions

^{V and} V,

defined by (3.2),

^are

both

viscosity

^solutions

of

^the

boundary

^value

problem (3.3),

^{with H}

defined by (3.4).

PROOF.

^We

^only

prove that ^V is a

viscosity

solution of

(3.3).

^The

proof

for V is

similar,

^and

we leave

^{it to the} reader. To see that V is a

supersolution

of

(3.3),

^let p ^E

Cl (0),

^y

E 0 and V* - ~p

attains its minimum at y. We may

assume

min(V* - p)

^{= 0}

by adding

^{a constant to p. We may also assume either}

n

y ^E fl, ^or y ^E an and

V* ( y) h(y),

^{because otherwise we have}

nothing

^to

prove.

We suppose

and will

get

^a contradiction.

By continuity,

^{there is a}

positive

^{constant e} &#x3E; 0 such that

Moreover ^{we assume}

and

Select M &#x3E; 0 so that

I g (x, a)

^for

(x, a)

^{x A. Note that, if}

then

X(s) = X(s; x, a)

^E

^{B(y, e),}

^{for 0 ~ s}

^2M.

^Set

t ⁼

E ,

and choose

2 a S

&#x3E; 0 so that

e(l - e - t )

&#x3E; 26.

By

^Lemma

^3.1,

^{for any}

x E

B (y, ±) ^nO,

^{there is an a = a ~ such that}

if if

Therefore,

^{if t r, then we have}

Also,

^if t &#x3E; _r, then ^we have

In both cases, ^we thus obtain

a contradiction.

Similarly

^{we can check} that V is a subsolution of

(3.3),

^{and we omit}

giving

^the

proof

^{here. D}

4. -

Continuity

^of

viscosity

^solutions

We continue to

study problem (HJ)’ - (BC)’

for Bellman

equations.

^Let

0, A, f, g

^{and h be as in}

§3,

^{and .H} be the function defined

by (3.4).

^We

present

a sufficient condition under which

(HJ)’ - (BC)’

^{has a} continuous

viscosity

solution. In order to state our

assumptions,

^{we need the notation: For a subset}

S we write for the closed convex hull of S.

Throughout

^this paper, the term "cone" will stand for a cone with vertex at the

origin.

^{For a subset S}

of R N and s &#x3E; 0, ^{we denote}

by cn, S

^{the e - convex conic}

neighbourhood

^of

S, ^{i.e. the set}

We note that

cneS

^{is a closed} convex cone We set

for

The

following

^are

pieces

^{of the} sufficient condition mentioned above. Let z Ean and A c aQ .

(A2)

^{There is an} open ^convex cone K and a constant e &#x3E; 0 for which

G (z)

^{c K}

and

(z

⁺

K)

ⁿ

B(z,,6)

^{n H = 0.}

(A3)

^{There is an} open ^convex cone K and a constant c &#x3E;

0

^{such that}

G(z) nK=l 0

^and

(x+K)

^{c fl, for xc-}

B(z,,E) ^n?i.

(A4)

^{There is an} open convex cone K and a constant e &#x3E; 0 for which for and

(

for

(A5)

^{There is an} open ^convex cone K and a constant c &#x3E; 0 such that

G(z)

^and

(z+K)

^{nil =}

^0.

It is _easy to check that if z is an interior

point (relative

^{to of} A, ^then condition

(A4)

^is

equivalent

^to

(A4)’

^{There is an} open ^convex cone K and a

constant

^e &#x3E; 0 such that

G(z)

^{c K}

and

(~ + ~) nB(z,e)

^c

^0,

^{for x E}

B(z,e) ^{n n.}

THEOREM 4.1. Assume

(A 1 ).

^{Let r and E} be,

respectively,

^an open and

a closed subsets

of

^ao such that ao ⁼

Let fo

^{be an}

open subset

of

^r. Assume that

(A2), (A3), (A4),

^{with A =}

ro,

^and

(A5) hold, respectively, for

^{z E}

E,

^{z E}

r,

^{z E}

ro

^{and z E}

rBr o. If

^{u is a}

viscosity

^solution

of (HJ)’ - (BC)’,

^{with H}

defined ^{by (3.4),}

^{then u E}

^C(fl),

^and

^ulll

^{has a}

^unique

continuous extension

to a

which is also a

viscosity

^solution

of (HJ)’ - (BC)’.

It may

happen

^{that some of}

F, E

^and

Fo

^are empty ⁱⁿ this theorem.

Proposition

^{1.3 or Theorem 3.1}

guarantees

^{the existence of a}

viscosity

^solution

of

(HJ)’ - (BC)’.

^{Thus we see, from this} theorem,

that under

^{the above}

^hypotheses

(HJ)’ - (BC)’

^{has a}

unique viscosity

^{solution in}

C (0).

The

assumption

^{of Theorem 4.1 is}

motivated ^by

^the

^following

observation.

Let T &#x3E; 0, and

set li = (0, T) x fi.

^{Let E = =}

(0,

^and

ro = {T}

^{x Q.}

^{Clearly E}

^{and r are,}

respectively,

^a closed and an open subsets of 9H and

ro

^{is an} open subset of F. Let

f and g

^{be as above and}

satisfy (Al).

^{Let A} E R, and define

f :

^{x A - R and}

g :

^{x A -}

by

=

and g(t, x, a) = (-1, g(x, a)),

^for

(t,

^x,

a)

^{E x A.}

The associated Hamiltonian

given by

for t,

s E R and _{x, p} and the

corresponding

^Bellman

equation

^is:

On the other

hand,

^if

(A3)

^and

(A5)

^{hold for z E} afi, ^then

(A2), ^(A3),

(A4),

^{with A =}

Fo,

^and

(A5) hold, respectively,

^{for z E}

E, z

^E

F, z

^E

fo

^and

z E

fBfo.

^This observation and Theorem 4.1

imply

^{that if}

(A3)

^and

(A5)

^hold

for z E ao and h is a

given

continuous function

ona fi,

^{then the}

problem

in

on

has a

unique viscosity

^{solution in where Note}

that this

problem

^{for u is}

equivalent

^{to the}

problem

in

on

for where Also note

u C .r.

that the

following

theorem and Lemmas 4.1 and 4.2

assert

^that the solution of this

problem

^{does not}

depend

^{on the values of h on}

Fo

^and

equals

^{to h on ~.}

Thus our result covers the

initial-boundary

^value

problem

for Hamilton - Jacobi

equations.

THEOREM 4.2. Assume the

hypotheses of

Theorem 4.1. Let u and v

be, respectively, viscosity

^{sub- and}

supersolutions of (HJ)’ - (BC)’,

^{with H}

defined by (3.4).

^Then

on

Moreover, (ull1

^and

(vIOBfo,O)*

^are,

respectively, viscosity

^sub-

and

supersolutions of (HJ)’ - (BC)’.

For the time

being,

^{we assume} Theorem 4.2 is correct and _prove Theorem 4.1.

PROOF. Let u be a

viscosity

solution of

(HJ)’ - (BC)’. By

^{virtue of Theorem}

4.2 we have

on

while on

ii, by

definition. Therefore

on

Thus is continuous

on n

and is a

viscosity

solution of

(HJ)’

-

(BC)’, by

^{Theorem 4.2. The}

uniqueness

^follows

immediately

from Theorem

2.1. D

In what follows we

always

^{assume the}

hypotheses

^of Theorem 4.2 and will prove the theorem.

LEMMA 4.1. One has

for

PROOF. We

begin by observing

^that

where ,

Fix z E

By translation,

^we may ^{assume z = 0.} Since

by assumptions (A2)

^and

(A5)

^{there is an} open ^convex cone K and an e E

(0,1)

^{such that}

and

Choose finite sequences and so that

and Set and for

Note that .... -- for

By

^the

continuity

^of g,

we see

that e (x)

^E

^K,

^for

^x

^{in a}

neighbourhood

^{of 0. Set}

Cc =

x

E B(0, E) }, ^E,

⁼

^{cnE, CE}

^and

^Ka

⁼

^cneEe.

^Then

^{CE , Ee}

^and

^Ke

^{are nonempty}

closed convex cones.

Replacing

^e &#x3E; 0

by

^{a smaller one, we} may ^assume that

and

for

Define d :

R N -+ 0, oo ) ^by

^{= dist}

( x, E~ ) ,

^{for x E}

^{JR N.}

^{It is}

well-known that d e and

for ~ 1

where

P ~ x)

denotes the nearest

point

^of

E,

^{from x.}

(One

way ^{to see}

these,

^for

instance,

consists in

observing

^that

and then

applying general

^results

(e.g.,

^{H. Br6zis}

[3, Prop. 2.11])

^{for convex}

functions to

d2 ).

Note also that d is

positively homogeneous

^of

degree

^1.

We claim that

for

In view of

(4.4), by continuity,

^{it is}

enough

^to prove

(4.6)

^{for x E}

K~ .

^To

this

end,

^{recall that and set then}

Assume x E

Kc

^{and that}

~P(x)~ &#x3E; ,8lxl.

^Then

As and we find that ^, The

homogeneity

^{of d}

yields that

. ~

Thus we have

proved (4.6).

Let x E

E~

^and p ^E

Ca satisfy Ipl

^{= 1.}

^{Let q}

^E

B ( o,1 ) .

^{Since p + eq E}

^E,

and

P(x) ~ (P(x) - x)

^{= 0, we have}

Taking q

^{= with x E}

E~ ,

^{in this}

inequality,

^{we find that}

for and

Now fix M &#x3E; 0, ^{so that}

Let a &#x3E; 0, and select a constant C &#x3E; 0 so that

Let _p E

E~,

^{and set for with}

to be fixed later on. Since we have for

and

hence, by (4.4),

By

^{the same reason, we have}

for

Therefore, using (4.6),

^{we see} that if B &#x3E;

2 max 1!2, ~ } ,

then w &#x3E; h on

B (0, e)

ⁿ an and w &#x3E; u* on

aB(0, e) ^nn. Using (4.7)

^and

(4.5),

^{we calculate}

that

for Fix

and set

no

^{= n n}

B(0, e)°

^and

ho (x)

⁼

w (x),

^{for x E}

^(9f]o.

^{Then z,v satisfies}

in

on

in the classical sense, while u* is a

viscosity

subsolution of

(HJ)’ - (BC)’,

^with

00

^and

ho

ⁱⁿ

place

^{of n and}

h.

^A

^simple

^argument which leads to a contradiction

now shows that u *

on 00 and,

ⁱⁿ

particular, u*(0)

^{+ a +}

B d (- p).

Sending

p ^- 0, ^while

keeping

p ^E

E~,

^{and then a}

10,

^{we conclude}

(4.2).

D LEMMA 4.2. One has

for

Our

proof

is similar to the above one, and we

only give

its outline here.

OUTLINE OF PROOF. We may ^{assume z = 0.} Fix e &#x3E; 0. We set and

By assumption ^(A2),

^{we can choose an e} &#x3E; 0 so that

and

for

Let and

B &#x3E; 0,

and define

by

There is a

positive

^constant

Ba , independent

^{of p,} such that if B &#x3E;

Ba ,

then w

^{h on}

B(0,e)n80,

^{z,v v*}

on 8B(0,e)nn

^{and 0}

in in the

classical

^{sense. Thus we}

^conclude,

^as in the above

proof,

that w v* on

B (0, e) n a,

^{and hence}

h(0) v* (0).

^D

In what

follows,

^{we write}

and

We note that u u * and v &#x3E; v * on an and that û ⁼ u * and v ⁼ v * in fi.

LEMMA 4.3. Assume

u* ( x ~ &#x3E; - C’ for

^x

E fi

and some constant C &#x3E; 0.

Then, for

^{each z E} r, ^{there is an} open convex cone K and a constant e &#x3E; 0 such that

and

for

PROOF. Let z E r. We may ^{assume z = 0.}

By assumption ^(A3),

^{there is}

an open ^convex cone K and a constant e E

(0,1)

^{such that}

G(0)

ⁿ

K ~ ~

^and

for We select finite _sequences

and so that

and Set

and for

Also,

^{set and define}

and

Replacing e

&#x3E; 0

by

^{a new one, we} may ^assume that for

and

for

In view of

(4.11),

^{it is}

enough

^{to show that}

(4.9)

^{holds for K =}

Feo.

^To

this

end,

^we will prove

that,

^for any ^{r E}

(0,e],

for where

and

Note that

(4.9),

^{with K =}

Fz,

^{is a} direct consequence of

(4.13). Also,

^notice that

Let r E

(0, e],

^{and let}

b, Mr

^{and B be as} above. Observe that for