Some remarks on quasi-variational inequalities and the associated impulsive control problem

A NNALES DE L ’I. H. P., SECTION C

B. P ERTHAME

Some remarks on quasi-variational inequalities and the associated impulsive control problem

Annales de l’I. H. P., section C, tome 2, n

^o

3 (1985), p. 237-260

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Some remarks

on

quasi-variational inequalities

and the associated impulsive ^control problem

B. PERTHAME

E. N. S., 45, ^rue d’Ulm, 75230 Paris, ^{Cedex 05}

Vol. 2, n° 3, 1985, p. 237-260. Analyse ^{non linéaire}

ABSTRACT. ^- We

study Quasi-Variational Inequalities:

In

general, (1)

^{has no}

solution,

^we prove here that

(1)

^{has a}

unique

^maxi-

mum subsolution that we caracterize. Then we compare the

implicit

obstacle

(2)

^{and the obstacle :}

and we

finally

^show

that,

^under

general assumptions,

^the solution of

(1)

is Holder continuous.

Key-words : Quasi-Variational Inequalities, Implicit obstacle, ^maximum subsolution,

Holder continuity, impulsive ^control.

RESUME. ^- Nous etudions les

Inequations Quasi-Variationnelles :

Annales de l’lnstitut Henri Poincaré - Analyse ^non linéaire - Vol. 2, 0294-1449 85/03/237/24/S 4.30/(E) Gauthier-Villars

En

general (1)

^n’a pas de

solution,

^{nous montrons} ici que

(1)

^{admet une}

unique

sous-solution maximale _que nous caracterisons. Nous comparons ensuite l’obstacle

implicite (2)

^et l’obstacle :

et nous finissons par ^montrer que, ^sous des

hypotheses generales,

^{la solu-}

tion de

(1)

^est holderienne.

Mots-clefs : Inéquations Quasi-Varationnelles, ^obstacle implicite, sous-solution maxi- male, continuite Holderienne, ^controle impulsionnel.

We

study

^{here the}

Quasi-Variational Inequalities (Q.

^V.

I.)

^{and the}

associated stochastic

impulsive

^control

problem:

where Q is a

regular (C3)

^{connected open set of and M is}

given by :

The

problem (1)

^was introduced in A. Bensoussan and J. L. Lions

[1 ] (more

recent results _may be found in

[3 ]).

^A

typical

^{result is the}

following

^{if we}

assume ~p ^-

0, f >_-

^{0 and co}

increasing,

^then

(1)

^{has a}

unique

^solution

which is in

Here we will relax these

assumptions

^{and answer the}

following questions.

If we take

general

~p

and f, (1)

^{has in}

general

^no solution because of ^a diffi-

culty involving

^the

boundary

condition : there is no reason a

priori

^{that Mu}

should be above _~p on aQ. And if in

general

^{there is no} solution of

(1),

the

question

^{is to determine in which sense}

(1) might

^{be solved} and whether the

corresponding

solution is the

optimal

^{cost function} for the associated

impulse

^control

problem

^{which is}

meaningful

without any condition.

If the solution of

(1)

^{is not}

continuous,

^{then formula}

(2)

^{has no clear}

meaning

^{and one}

generally

defines the

implicit

^obstacle

by:

But it is not clear

(and

^{to our}

knowledge

^{it has never been checked before}

!)

that Mu ⁼

M + u (at

^{least for u E}

C(Q)).

^{We also answer here that}

question.

The last

question

^{that we}

study

^{is to find}

general

^local

regularity ^results,

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

say in

C°~°‘,

^{which do not} involve the

boundary

^{data Such a}

regularity

has

already

^been

proved by

^J. Frehse and U. Mosco

[9 ].

^{Here we state}

an

analogous

result but our

proof

^is

completely

^{different from the one in}

[9 ].

In a first

part

^{we show that}

(1)

^{has a}

unique

^maximal subsolution which is the solution of:

Here we cannot look after solutions in

H~

since this space is ^not

adapted

to the obstacle Mu

and,

ⁱⁿ

particular,

qJ A

Mu ~ H1/2(~03A9).

^{But under a}

general assumption

introduced in B. Perthame

[16],

qJ A Mu and Mu

are continuous so that we can deal with continuous solution of

(3)

^called

viscosity

^solution. This kind of solution

(a particular

^case of the notion introduced in M. G. Crandall and P. L. Lions

[7],

P. L. Lions

[13 ] [14 ])

allows us to solve

(3)

^{with the classical}

argument

of B. Hanouzet and J. L.

Joly [10 ].

^{Remark that in}

Appendix

^{2 we} prove

equivalence

^between

different notions of solution of the obstacle

problem.

^In

particular

it appears that the

viscosity

solution is also the classical solution in

Hloc.

We prove in ^a second

part

^{that this}

analytical

^{solution is the one we}

should

expect

^{in view} of the associated

impulsive

^control

problem.

^This

is a verification

analogous

^{to the one in}

[3 ] [17] which clearly

^{shows that}

the above notion is the correct one.

We

give

^{in a third}

part

^some

comparison

^{results on Mu and}

M + u.

There are two main results : ifQis

sufficiently

^smooth

(CN)

^then

^{Mu = M + u}

a. e. and if Mu or

M + u

is continuous then Mu ⁼

M + u everywhere (remark

Mu and

M + u

^{are defined}

pointwise

^{and not almost}

everywhere).

Finally

^we prove that when _co is continuous and a small

enough,

^{then the} solution of

(1)

^{is in}

C°~ .

The method is to

replace

^Mu

by

^an other obstacle which is in and which is built

locally

^{with the}

help

of

general properties

^oi iwiu. Remark that we can

give

^an

example

^with

very

regular

data where u is

only (cf.

^{B. Perthame}

[18 ]),

^{and that}

in

general

^no

regularity

up ^to the

boundary

^holds.

I.

Q. V. I.

WITHOUT THE EXISTENCE OF A SUBSOLUTION

1.

Assumptions

and main results.

This section is devoted ^to the

Q.

^{V. 1.:}

Vol.

We prove

(1)

^{has a}

unique

^maximum subsolution and that it is the solu- tion of:

,

Here we call subsolution of

(1)

any u ^E

C(Q), satisfying :

where we have set :

We assume :

where k is ^a

positive

constant, co :

(f~+)N

^{-+ is a lower} semi-continuous sub-additive function with

co(0)

^{= 0}

and ~ >_

⁰

means ~ = (~ 1, ... , ~N)

0. I

We will also assume that the

boundary

data satisfies :

These

assumptions

will allow us to deal with continuous solutions of

(1)

or

(3)

^{which are called}

viscosity

solutions. Recall that P. L. Lions

[13 ] [14]

has introduced the notion of

viscosity

solution of

general

second order

equations adapting

^{to second order}

equations

^the notion introduced

by

M. G. Grandall and P. L. Lions

[7]

for first order

equations.

^{A very}

parti-

cular

application

^{of this notion}

yield

^{that the}

following

^obstacle

problem :

has a

unique viscosity

^{solution u E}

C(Q)

^{if the obstacle ~I’ E}

C(Q).

^{In this}

particular setting

^{u is}

nothing

else that the limit in

C(Q)

^{of all}

regularized

obstacle

problems

with nice standard solutions.

(The

different notions of solution of

(10)

^{are collected in}

Appendix

^{2 where we} prove the

equivalence

of these

notions).

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

In the

following

^{we will call}

viscosity solution

^of

⁽³⁾

^a continuous func- tion ^u such that ^A

Mu,

^{Mu E}

C(Q)

^{and u is a}

viscosity

^solution

of

(10)

^{with T = Mu.}

Remarks.

1)

Below it will be

proved that,

^under

assumption (9)

^and

with the definition of Mu in

(2),

^we

always

^{have Mu E}

C(Q)

^{when u E}

C(Q)

and A Mu.

2)

^{In the}

particular

^{case treated}

here,

the notion of

viscosity

^solution

is

equivalent

^{to and :}

moreover in

Appendix

^{2 it is}

proved

^{that M E}

H ~ .

3) Exactly

^as in B. Perthame

[16 where

^the

assumption

^has been intro- duced to

get

^the

continuity

^{of the}

solution,

^one

easily

^{checks that}

(9)

may be

replaced by :

where w is any continuous _super solution

of ( 1 )

^{such that w} ~p . Then we can state the :

THEOREM 1. ^- Under

assumptions (5)-(9), equation (1)

^{has a}

unique

maximal subsolution u and it is the

unique

^solution

of (3).

If

03C6 ^E and

M003C6

^E

(resp.

^is

locally semi-concave)

^then

u E

(resp.

Let us recall that a function u E is said to be

locally

semi-concave if for each _open set U~ ^c U~ ^c S2 there exist a constant C such that :

or, in other words: is concave on convex subsets of O.

The end of this section is devoted to the

proof

^of

^Theorem 1,

it is divided in three

parts :

^{first we build a}

decreasing

process that is

uniformly converging

to a solution of

(3) (this

^{is a variant of the « usual »}

proof

^{due to} B. Hanouzet and J. L.

Joly [10 ]).

^{Here the}

difficulty

^comes from the fact that the

boundary

value of the _process

changes

^{at each}

step.

^The

concavity

^{of the}

operator

which associates _~p A Mu to u enables us to conclude. In the second

part

we state the

uniqueness

^result

^and finally

^we prove the

regularity.

Vol.

2. Existence of a solution.

Throughout

^{section I we will assume that}

f >__ 0,

qJ ^> 0. Indeed

(7)

enables us to make such an

assumption by adding

constants to u and _q

. and a

positive

^function to

f

^{We can now} define the

following

process : uo e

C(Q)

^{is the} solution of:

r A ,

then we define u 1 as the solution of:

and

by induction,

The existence of this sequence is

justified by

^the:

LEMMA 1. ^- For

each n >_

⁰

Mun

^E

Proof

^{It is}

easily

^checked

(cf. [16 ])

^{that under}

assumption (9) Muo

is continuous on Q.

So,

^{we assume}

by

induction that

for n >_ 0, Mun

^is

continuous and we prove that

Mun + 1

is also continuous. We know

(see [l6 ])

that

Mun+ 1

^{is lower} semi-continuous. Let us prove it is upper semi-conti-

nuous.

j

Let : ^I

~o ~ 4;

1 if

then,

on a

neighbourhood of xo

^{we have :}

and the result is

proved.

^{If xo + we} will prove that :

and this prove that

Mun+ 1 is

upper semi-continuous at xo since :

But

(12)

is deduced from the:

LEMMA 2. ^-

0, then

v(xo + ço)

⁼

Mv(xo + ~ o) -

^k.

’

Indeed take v ⁼ in lemma

2,

^then:

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

since un is ^a

decreasing sequence).

^But xo

+ ~

^{E r and thus we have}

and

(12)

^is

proved, concluding

^the

proof

^{of Lemma 1.}

Proof of

^{Lemma 2. - Notice}

that,

^if r~

> 0,

xo +

~o

+ ~ ^E

5,

then :

and, taking

the infimum over

0,

^{we find :}

since

equality

^holds

for 11

⁼

0,

^{Lemma 2 is}

proved.

Remarks.

1)

^This also proves Remark 1 in I .1: take un+ 1 ^_-- un ^{--_ u} in the

proof

of Lemma 1.

2)

^The

argument

used above is very similar ^to the one introduced in L.

Caffarelli and A. Friedman

[5] [6 ].

As mentionned in the

proof

^{of Lemma}

1,

the sequence Un is

decreasing,

0 since

f >_ 0,

qJ

> 0,

^{so that} un converges ^{to some} function u;

moreover we have :

PROPOSITION 1. ^- The sequence un converges

uniformly

^{to u E}

C(SZ)

which is a

viscosity

^solution

of (3)

^{and 0 _ u} uo.

Proof

^{- The}

proof

^{below is}

adapted

^from B. Hanouzet and J. L.

Joly [10].

Notice that

(3)

^{is for}

example

^{deduced from the «}

stability »

^of

viscosity

solutions of second order

equations by

^uniform convergence. Let ^us prove

the uniform convergence of _n

choose p

^e

]0,1 [

^{such that}

-

^,

k ^~

~ ~u0~L~

^{and assume that for some 0 e}

[o, 1 ] and some n >__

^{0 we have :}

then,

^{since M is a concave}

mapping :

Let us call z the solution of the obstacle

problem (10)

^with

and _vo the solution of

then, applying

the maximum

principle (regularizing ^Mun

^if

necessary)

and

noticing

^that:

one

easily

checks that :

But the choice

of p

shows that: _vo and thus we obtain:

un + 1 - Un+ 2

C ~(1 -

^{1 .}

Since _{uo - uo,}

iterating

^{the above}

inequalities

^{we obtain:}

and this proves

Proposition

^1.

Remark. ^- Of course this also proves that

Mun

converges

uniformly

to Mu hence u is a

viscosity

^{solution of}

(3).

3.

Uniqueness.

We must prove ^two results : that the solution of

(3)

^is

unique

^{and also}

that it is the maximum subsolution. This last

point

^is

clear,

^{if a function}

w E C(S2)

satisfies :

(this actually

^{means that w is a}

viscosity

subsolution of

(1)

^{if Mw E}

C(Q)) then, recalling

^{the result of} P. L. Lions

[13 ], w

uo since for

example

^w

is ^a

viscosity

subsolution of

(11).

^We deduce that:

and so

w

u 1. An easy induction proves that

w

un for

each n >__

^{0 and}

PROPOSITION 2. ^- Under the

assumptions of

^theorem

l, if

^{w E}

is a

viscosity

^solution

of (3),

^{then w = u.}

Proof

^{- We}

already

^{know that}

^{w _}

^{u. First we show that ~w >_ 0:}

we know

(by

^{the remark in I}

2°)

^{that Mw E}

C(Q)

^{so we can build a}

sequence

E such that Mw in

C(Q).

^{Let us denote}

by wn

^{the solu-}

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

- tion of

(10)

^{for T = so that} ~ ~~ ^w in

C(Q)

^and

(see [16]).

^Take

^{xn ~ 03A9}

^so that : Min ⁼ as

xeQ

we may ^assume

(at

least if n is

large enough)

^that

wn(xn) Mwn(xn).

If xn ~ 0393

^{we deduce that}

03C6(xn) ~ ^0,

^{if not} the maximum

principle

shows that

>

^{0 and in both cases 0 and thus w > 0.}

We ^now prove

Proposition

^{2 with the}

help

^{of B.} Hanouzet and J. L.

Joly

method. We have :

by

^{the same}

^arguments

^{as in}

Proposition

^{1 we}

get

and

^so

u __

^wand

Proposition

^{2 is}

proved.

4.

Regularity.

Here we assume that ^E n

C(Q) (Resp.

^that

M003C6

^is

locally semi-concave)

^{and we} prove that u ^E

(resp.

We

only

^{sketch the}

proof

^{since it is}

nearly

^{the same as in}

[16].

^One

introduces :

and one

easily

proves that there exists some open ^set G

(for

^the

topology

of

Q)

such that :

~

0

~

0

and so u E

(resp.

^{u E}

Now,

^{as Mu « takes} its values » in F and with the

regularity

^of

Mocp

^{one can} prove that Mu is in

(resp. locally semi-concave)

^and

applying

^{classical results on the obstacle}

problem

this proves that ^{u E}

(Resp.

^{u E and the}

proof

of Theorem 1 is

completed.

Remarks.

1 )

^In

particular

^{if Q is convex the solution}

belongs

^to

, when _~p E Indeed

Muo

^E

C(Q)

^{and is}

locally

semi-concave and this is

enough by

remark 3 in 1.1

(cf. [1~ ], [18 ]).

2)

^{Of course these} results extend to Hamilton-Jacobi-Bellman _equa- tions

(see

P. L. Lions

[12 ],

L. C. Evans and P. L. Lions

[8 ]).

^{The associated}

Q. V. I.

^{is :}

(For

^{results on} this

problem

^see S. Lehnart

[11 ],

B. Perthame

[16 ]).

Vol.

II. THE IMPULSIVE CONTROL PROBLEM

Our purpose,

here,

^{is to}

give

the stochastic

interpretation

^{of u, the solu-}

tion of

(3),

^{in terms of} control of diffusion processes. Two remarks are to be

developped

^{below :}

first, despite

^{of the}

jumps

^the process is constrained

to

stay

ⁱⁿ

Q;

next, the

change

^of

boundary

^data

(from

qJ

to qJ

^A

Mu)

^does

not induce _any

change

^{on the}

optimal

^{cost function.}

1. The

optimal

^{cost function.} _

Let

(Q, F, Ft, P, Wt)

^{be a standard} space

composed by

^a

probability

space

(Q, F, P)

^{with a}

right-continuous increasing

filtration of

complete

sub-a

algebra Ft

^{and a} Browian motion _wr in

Ft adapted.

For any sequences

~

¹

G 2

^... C~n

... , C~n

_{n ~} oo, of

stopping

^{times and}

03BE1,

^...,

03BEn,

^...

of FOn-random

^{variables in}

(R+)n

^{we can define}

by

induction :

and :

we set :

We will say that such ^a

system

^~ is admissible if E Q whenever Then we define

and the cost function of this

system

^is:

and the

optimal

^cost function is:

Remark.

1)

^{With our} definition of admissible

systems

^{we know that} and so

(15)

^is

meaningful.

2)

^{Here we have taken}

c(x) _

^{~, to}

simplify

^{notations but the}

following

results still hold with any

c(x)

^{> 0.}

THEOREM 2. ^- Under the

assumptions of

^Theorem

1,

the solution u

of(3)

^{is the}

optimal

^cast

function given by (16)

Annales de l’Institut Henri Poincaré - Analyse ^non linéaire

Proof 2014 z) u _

^{v :} This is clear in view of A. Bensoussan and J. L. Lions

[4]

or P. L. Lions

[.l3 ] :

^u is solution of the

Q. V. I. (1) with cp

^{= qJ .A Mu}

so that it is the

optimal

^{cost function for a cost} function defined

by (15) with

ⁱⁿ

place

^of qJ. These cost functions are less

than j(x, A)

^since

(p

qJ

so that u _ v.

ii ) u >__

^{v : Here we}

give

^an

optimal

admissible

system

^{as in}

[3 ] [I 7 ].

Let ~

be measurable function in such that :

we can find a standard _space

(Q, F, Ft, P,

^{where we solve :}

let

C~1

be the first time when

y° belongs to ~ _ ~

^{u =}

Mu ~ (%~

^{1 = + oc}

ifrø is not

reached), ~

^{1 =}

(1

^is

anything

measurable if

C~1

^{= +}

oo )

and

by

^induction

’

and

G~n + 1

is the first

time yl

^{enters ~ and}

~~‘ + 1

⁼

1 )~.

From

[2 ] we

^{know the :}

LEMMA 3. ^- For any n

> 0,

^{one has :}

Remarks. 1

°)

^and

yx(i)

^{are in the}

set (

^{u Mu}

~,

^{indeed whenever}

reaches {u

⁼

Mu ~

^it

jumps

^{to come back}

in { u Mu ~.

^In

^parti-

cular

u( yx(2)) 10 ~ ~ _

^{oo ) .} _

2°)

^{One checks as in}

[~] ]

^that n ~ ^{+ oo ;}

indeed

for

each n,

u( yX-1(~n)) _

⁺

~n) > k, ^and,

^{since u is}

continuous,

^{>_ L > o.}

Since n ~

^{0 we}

obtain 1

^{+ ...}

+ n ~ Ln

^{and the formula:}

on

{~ oo }

^{shows that}

^~

m + ^oo.

In

(17)

^{we remark that :}

but we have :

since

Un __ ~

^{whenever oo. Thus}

adding (17)

^for

each n >_

^{0 and}

using

remark

2,

^we

get

and this means that

u >__

^{w and} theorem 2 is

proved.

III. SOME REMARKS ON THE OPERATOR M

The

operator

^{M which} defines the

Q.

^{V. I. is not}

always given by (2).

Indeed when one deals with discontinuous solution the

operator

^Mu has ^no clear

meaning

^{and the}

Q.

^{V. I. is}

generally

defined with ^an ope- rator

M + :

From a stochastical

viewpoint

^it is also natural to consider admissible

systems

such that the

jumps ç

^{at a}

point x

^are constrained to

satisfy x ^{+ ~}

^{E Q}

and ^no

longer

It will rise ^an other

operator,

defined for x E 12:

M +

is defined _{for any} bounded from below measurable function and M- for u E

C(Q).

^{Here we}

give

the relation between

M, M -,

^{M + .}

1.

Regularity

^of

M, M-, M+.

PROPOSITION 3. ^- Let u E

C(Q)

^{then Mu is lower} semi-continuous on ~

(I.s.c.),

^{M _ u is} upper semi-continuous

(u.

^s.

c.)

^{and Mu _ M -u on S2.}

Proof

^The

regularity

^{of Mu and the}

inequality

^are clear. The other results are based on the

following

^{remark: take un E}

C(Q)

^a

decreasing

sequence such that u,~ n ~ ^u

pointwise

^{then decreases to M _ u.}

Now if we choose _un such that

un > ~ ~

^{u and} un

‘~

on

Q,

then it is

easily

^{checked that} is continuous and _so, that Mu is ^{u. s.} c.

PROPOSITION 4. ^-

If

^{u is u. s. c.} then M-u is u. s. c. moreover,

i~

^{co is}

continuous and u is u. s. c. then M -u ⁼ M _{+ u} on Q.

Proof

^The

proof

^of

Proposition

³

directly

proves the first

part

^of

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}

Proposition

^4.

Moreover,

^if co is continuous

and un

is chosen as above it is clear that

M + Un

^{= so that:}

M _ u _ M + u _ M+un

_{n ~}

M + u

^on

Q,

^and

Proposition

^{4 is}

proved.

2. Relation between M and M-

(or M + ).

PROPOSITION 5.

- If Co is continuous,

^{u E and Q is}

of

^class

^CN

then Mu ⁼

M + u

^{a. e.}

Proof of Proposition

^{5. Let us introduce some} notations : F will be the set of

points

^where

possibly M + u(x),

^{and will denote}

by Tr( y)

^the

tangent hyperplane

^{to ~SZ at} y and

v( y)

^the unit outward normal to aSZ at y. Then we have :

F ^c

G== { ~- EO, 3y

> x, y ^E

(ei, y-x)=O

^{for some i and x E}

where is the canonical basis of

[RN.

This result has been

proved

in

[15 ],

^{but it is not}

precise enough.

^{In fact we have}

F c ^Gn

^{where :}

nN

Gi 1 - ~ x E ~,

^y

> ~ E)~

^N such that :

.

Let us prove that

F c Gn :

^{take some xo E}

^F,

^{as xo E} G there exists

nN ~

y ^{> x,}

y ~ ~03A9

^{and some} eil such that

(etl, y - x)

⁼ 0. We may

always

choose ei2 ^... eik ^... ein such that : for 1

__ j k, y - x) ~

⁰

for j >

^{k. Now assume that it means that for some 1}

> k, (eil, v( y)) ~

^0.

Since _y > x then _y + ^s _e; ^{> x} for E small

enough,

^and either y - E eil ^E

Q,

or

y +

^E eiz ^{E Q} for a small

enough.

^{In both cases we}

get

^that

M+u(xo)

⁼

Mu(xo),

indeed we know that

M + u(xo) >_- Mu(xo)

^and

This

yields

^a contradiction and thus

x E Gk.

^{We now conclude with the :}

LEMMA 4. ^- For 1

_ k _ N,

^meas

(Gk)

^{= 0.}

STEP 1. ^- We _prove that

Gi

^{is of} zero measure and it is

enough

^to

prove that

Vol.

v( y)

^is

parallel

^{to eN and}

(eN,

y -

x)

⁼

0 ~

^{is of zero mesure. Since the}

set of _y E a~ such that

v(y)

^is

parallel

^{to eN is}

compact,

^{it can be covered}

by

^a finite number of open ^sets

where fi

^E

^CN and Oi

^{is an} open ^set.

Denoting y’ = ( y 1

^...

YN - 1)

it remains ^to prove that for these func- the sets : ^.

is of zero measure. But :

Denoting by

^the

Lebesgue

measure on R we

have, applying

^Sard’s

Theorem:

thus H is of zero measure and we conclude the first

step:

^meas

(Gi)

^{= 0.}

STEP 2. ^- In the same way ^we prove that

Gk

^{is of zero measure for}

k > 1: it is

enough

^to prove that

Hk

^{is of} zero measure where :

By

^the

implicit

function Theorem and

changing eN

^{into some N - k}

+ 1,

if necessary ^we have

locally :

and it is

enough

^to

study

^{the set:}

But its measure is less than :

... ,

yN - k + 1 ~~

this last term is the measure of the critical values of the function

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

and so is zero

by

Sard’s theorem. This concludes the

proof

^{of Lemma 4}

and of

Proposition

^5.

It is

quite

^{easier to}

get

^the

following comparison Proposition

^{in which}

we consider M-u and Mu ^are defined for x E 0:

PROPOSITION 6. ^-

If

^{u E}

C(S~),

^and

co is a non-negative,

sub-additive

function

^{such that}

co(o)

^--_

0,

^{then the set}

of discontinuity points

^{in 03A9}

of M-u

is also the set

ofdiscontinuity points in

^Q

ofMu

^{and it is also the set}

of points

where

M _ u(x) ~ Mu(x).

COROLLARY. ^-

lf u

^E

C(Q),

^{M -u = Mu a. e. in} Baire Sense.

This is because we know that the set of

discontinuity

^{of a 1. s. c.}

(or

^{u. s.}

c.)

function is rare in Baire sense.

Proposition

^{6 is}

clearly

deduced from the

following :

LEMMA 5. ^- Under the

assumptions of Proposition

^6:

Let us prove the second

equality

^{which is the most difficult one : we know}

that lim _sup and we shall prove

that:

Take xn

^> xp and _{xn xo,} +

CO(çn)

^{+ + 0.}

Extracting

^a

subsequence

if necessary ^we

assume: 03BEn

_n~~

03BE.

^Since xn +

03BEn -

^xp

^>

^{xp >}

0,

^{we can find an} EnE En n~~

0,

x0 + 03BEn + En

>

⁰

and We have then :

Remark.

2014 1)

^{Here we have used the three}

properties

^of co : co is subaddi-

tive, co( ç) _.~t

^{0 and co is 1. s. c.} To prove the first

equality

^of

Proposition

⁶

we

only

^need co ^to be 1. s. c.

2)

A useful consequence of

Proposition

6 is that the solution of

(3)

^is

also the solution of:

in other words u is also the solution of the

impulsive

^control

problem

where the admissible

systems

^are defined in such away that the process

yx(t ) stays

^{in Q}

(and

^no

longer

ⁱⁿ

Q)

^{and the}

stopping

time’! is the first

Vol. 3-1985.

exit time from Q

(and

^no

longer

^from

Q).

^{This remark also}

applies

^{to the}

results

proved

ⁱⁿ

[16 ] for Q. V.

I. associated to Hamilton-Jacobi-Bellman

equations.

IV. HOLDER CONTINUITY OF THE SOLUTION OF

Q.

^{V. I.}

In this section we prove ^a

general

^Holder

continuity

^{result for the}

Q.

^{V. I. :}

where

f >_

^{0 and}

a( . , . )

is the bilinear form associated with A :

(Au, v)

⁼

a(u, v)

Indeed if we do not assume

(9)

^the

general theory developped

^{in A. Ben-}

soussan and J. L. Lions

[3 ] shows

^that

(18)

^{has a}

unique

solution in

Ho,

but it is not continuous in

general

^{since our}

assumptions

^on co

(see (2))

are too weak. Here we

give

^a

proof

^{of the Holder}

continuity

^{of u under}

assumption :

^,

(19)

co is continuous on

((f$+)N, co(~) 0,

where cx is an

exponent

^to

be precised

in Theorems 3 and 4. Such ^an

assumption

^has

already

been introduced in A.

Bensoussan,

^{J. Frehse and}

U. Mosco

[4 ],

^or J. Frehse and U. Mosco

[9] ] to

prove the

continuity

^of

Q.

^{V. I. with}

quadratic growth.

^The method used here is

completly

^different

of the one in

[4 ] [9 ].

^{It consists in}

interpreting

^{u as the} solution of an obstacle

problem

^{where the} obstacle T has some

special property. Using

^a

regularity

result for linear second order

elliptic equations

^{on open sets}

satisfying

^an

exterior cone condition we prove that the solution is Holder continuous.

The

precise

result is the

following:

THEOREM 3. ^- Under

assumptions (5)-(7)

^{there exist some ao, 0} ao

1,

such that

if (19)

^{holds with 0 a} ao then the solution

of (18) belongs

^to

C0,03B1loc(03A9).

The

proof

^of this Theorem is divided in two

parts :

^{first we} prove that the solution u of

(18)

^{is u. s. c., then we} prove the Holder

continuity

^{of u.}

Remark. Of course this result extends to the more

general

^case

where f

is

eventually negative

^but

(18)

^{has a} subsolution. It also extends to any

boundary

^data.

1. Selection of an u. s. c. solution.

Here we make the

assumptions

of Theorem 3. We know that u is obtained

as the limit of the

decreasing

process ^as in I: uo is the solution of

Auo

⁼

f,

uo e

C(Q). Then un

^{is the solution of:}

Annales de l’Institut Henri Poincaré - Analyse ^non linéaire

At each

step

^{we can select} a u. s. c. solution un. Indeed if _Un-1 is u. s. c.

then

M-Un-1

⁼

M+un-i by Proposition

^{5 since co is} continuous

(in particular

it doesn’t

depend

^{on the} choice of the u. s. c.

selection)

^and

we know is u. s. c. Now take a

decreasing

sequence of obstacles

03A8p ^MUn-1

^the solution uP of the obstacle

problem

^{for is continuous}

p~~

and decreases

with p

to un

(moreover

^uP

_p~~

un in

Ho by

^the

general theory

ⁱⁿ

[3 ])

^{so that we can select} an u. s. c. solution _{un at} each

step.

Finally

^{since u~ is}

decreasing

^{to u, u has} a u. s. c.

representant.

2. Holder

continuity

^{of u} in theorem 3.

Take a sequence _n ^u in

Q,

^{un E}

C(Q)

^{and assume =}

II un

^is

large enough

^{so that} is continuous and We prove in what follows that each

solution vn

^{of the obstacle}

problem

^{for T = M-un}

belongs

^to

C°~

^with uniform bounds on

compact

subsets of Q.

For _any

x0 ~ 03A9

^{we denote}

by d(xo)

^{the function}

d(xo)

^{= inf}

d(xo, y).

yeon

Since co is subadditive and

by (19)

^we have for all _y E

Q, y --

xo : ¹

In the

following

^we will denote

by:

and we define two open ^{sets :}

By

^the

appendix

^{1 if a is small}

enough ( a ~

in Theorem

4

^{we can}

solve:

~ ^{~ ~}

Vol. 2, ^{n° 3-1985.}

then we set :

The

Appendix

¹ shows that

vxo(y)

^{is in}

^C°°°‘

^{if a is small}

enough

^and

I)

vxo

Ilco,a; ~ D(xo),

^where

D(xo) only depends

^on

d(xo),

^{and so the obstacle T}

defined below is in

C°~

ⁿ

C(Q) :

and we conclude

by

^{the :}

LEMMA 6. ^- The solution vn

of the

^obstacle

problem for

the obstacle M -un is the solution

of :

Proof

- For any x e Q ^we have :

hence w is a subsolution of

(18)

^and

w

vn. On the other hand for _any

xo E Q ^we have :

This

implies

vxo,

Vxo

^{i. e.} vn ~F and then _w; and Lemma 6 is

proved.

Now we deduce Theorem 3 from Lemma 6: we notice that the Holder estimates on vn do not

depend

^{on un and since} vn converges to u, Theo-

rem 3 is

proved.

Annales de l’lnstitut Henri Poincaré - Analyse non linéaire

APPENDIX I

CONSTRUCTION OF A BARRIER FUNCTION

FOR AN OPEN SET SATISFYING THE EXTERIOR CONE CONDITION In section IV we need to solve linear elliptic second order equations for open ^sets satisfying only ^{the exterior cone} condition. The _purpose of this appendix ^{is to show how} to construct a barrier function in such cases (which ^{do not seem to be} classical). ^{The results below are}

due to K. Miller [15 where ^{one can} find another way ^to prove them.

In this section we are given ^an operator ^{A and a function} f satisfying assumptions (and notations) of section I. We denote B ⁼ ~~ ^{bi and} by

Cu

^{the cone:}

where ^r = I x j, ^{and we construct a} barrier function at the point ^{0 for} the open ^{set :} The existence of a barrier function and its regularity ^are given ^{in the :}

THEOREM 4. ^- Let ,u ^E [0, ¹ [ ^then there exist a E ]0, ¹ [ ^{such that} for ^{h E there}

is a function ^{v E n such that:}

moreover a and ~~ ^v ~~~.« only depends ^on Ro, p, A, f ^and ~~ ^h Proof ^- We may ^assume h(0) = ^{0 and we look for v} satisfying :

We will divide the proof ^{in three} parts : computation ^{of Av,} lower bound for Av, ^existence of g ^{and a.}

1. Computation ^{of Av.}

We assume ,u, ⁺ 1 ], ^{and we} compute:

Vol.

Introducing y

= - r~ ,

^{... ,} r2 ^,

1

- ~12

^{we get :}

2. A lower bound for Av.

We assume 0, g’ >__ 0, g" ^{0 and we} put ^{t =} [ - ^{1, 1} ], ^{then we} get :

now we put ^{x =} (x’, xN) ^{and 1 so that the} coefficient of ra-

2g’(t )

^is equal ^to

here C denotes different constants depending only ^on

(aij)

^{and we assume N > 2.}

Finally ^we get : _ _

It remains to find some g for which this expression ^is large enough.

3. Existence of g.

LEMMA 7. ^- For [0, 1 [ there exist a E ]0, 1 [ ^and

g~ E C°°( [- ,u, 1 ], ~+),

g~ >_ ^1,

g~ >-_ 0, g;~

>_ 0 such that :

Proof of ^{lemma 7. - We look for} such that : then (21) ^{becomes :}

Annales de l’Institut Henri Poincaré - Analyse ^{non linéaire}

and a straightforward study ^{of the} function between the brackets shows that, if K is large enough (depending on and it is positive. Choosing ^{a small} enough ^we get ^{lemma 7.}

We can now conclude the proof of Theorem 4; by lemma 7 and choosing g ^{= we have:}

Av > ^where C(,u) is the minimum in (21), C(,u) ^> 0 ; for C large enough (20)

is satisfied and the dependance ^of C~ ^{v on} ~,, A, f and ~~ ^{h is clear.}

Remark. ^- Of course for each (3, ⁰ j~ a, and h E

{f~N)

^{we can find a barrier}

function in

Co°~{C~;)_

Vol.

APPENDIX 2

DIFFERENT NOTIONS OF SOLUTION FOR THE OBSTACLE PROBLEM

Throughout this paper ^we have used the notion of viscosity solution for the obstacle

problem. ^An other definition is given ^{in a remark of Section I.} On the other hand we could

use the usual definition of variational inequalities. The purpose of this Appendix ^{is to}

prove the equivalence ^between these three notions. Namely, ^we have, with the notations and assumptions of section I and IV:

PROPOSITION 7. ^- Let u and ’P belong ^to C(Q) ^{then the} following ^are equivalent : i) ^{u is a} viscosity ^solution of:

iii ) ^{and u is a} solution of the variational inequality associated to ’P i. e. :

iv ) ^{u is the} limit of solutions of

all

regularized ^obstacle problem

with

^{nice standard}

solution (actually this limit holds in C(Q) ^and weakly ⁱⁿ for all Q’ ^c Q’ c Q). ^{In other} words if we consider _uE solution of:

with 03A8~ ~ rpE ^on aSZ, 03A8~ ~ ^{03A8 in} C(SZ) and 03C6~ ^~ qJ in then _uE ~ u in C(H) ^and weakly ⁱⁿ

Proof ^{- The} equivalence ^between (i) ^and (ii) ^{is a} simple consequence of the notion of viscosity ^{solution as defined in P. L. Lions} [13 ].

The

equivalence

^{between (i ) and (iv) is easily deduced from the} stability (under ^conver-

gence in C(Q) ^{of the notion of} viscosity solution and the existence results for (23) ^which

are proved ⁱⁿ [2 ] and [16 ]. ^{Remark that the} family (ue) is bounded in (and ^{thus the} weak convergence holds) ^{since we have:}

and for ~ ^E ~+(SZ) ^{and C} large enough:

Annales de l’lnstitut Henri Poincaré - Analyse ^{non linéaire}