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350

and

a{x)([-e,e)

) CdH

{x,0)

(3.17)

Conditions (3.16), (3.17) guarantee

by the

well

known Chow's Theorem that

x<y

^ 0

and, consequently, d(x;y)

< +oo for all x,y (see, for

example,

[5]).

Moreover,

d is a sub -

Riemannian metric

Carnot

Caratheodory type which compares

locally with

the

euclidean distance

—

y\ on \R

Ci\x-y\ <d(x;y)

< C

\x - y\i

(see [13]).

follows that, locally,

—¥

d(x;y)

is £

Holder continuous

and is a

natural

candidate

to be the

required solution

of the

eikonal equation. Because

of the

multivalued

character

of the

control system (3.14), several technical refinements

to the

standard

dy-

namic programming argument

(see, for

example,

[3]) are

needed

order

show that

viscosity solution

of the

eikonal equation (3.6).

refer

to [7] for a

detailed

proof.

Let

only observe here that

in the

present setting

the

function

d is not, in

general,

differentiable almost everywhere;

the

notion

viscosity solution seems therefore

to be

essential

interpret

d as a

solution

(3.6).

A large class

examples, which

relevant both from

a PDE and

control theory point

of view,

which

our

generalized Hopf formula

u(x,t)

[

(y)

t*>(^l

applies

provided

homogeneous Hamiltonians

of the

form

(x,p)

\a(x)p\

where

a(x)

satisfies conditions (3.16), (3.17) above.

An interesting particular case (here

N = 3 to

simplify notations)

(3.18)

Ho(x,

)

= J fa -

)

+ fa +

)

arising

connection with

the

second order subelliptic operator

A#i (the

Kohn Laplacian

the

Heisenberg group,

see

[14]); note that

the

differential inclusion (3.14) reduces

this case

to the

symmetric system

in the

well

known Brockett's example

nonlinear

control theory,

see [6].

Our Hopf formula

(3.9)

coincides

this case with

the one

recently found

for

this

example

Manfredi

Stroffolini

[12].

References

[1]

ALVAREZ,

E.N.

BARRON,

ISHII, Hopf formulas

for

semicontinuous data, Indiana

University Mathematical Journal, Vol.

48, No. 3

(1999).

351

[2] M. BARDI, L.C. EVANS, On Hopf's formulas for solutions of Hamilton - Jacobi equations,

Nonlinear Anal., TMA 8 (1984).

[3] M. BARDI, I. CAPUZZO DOLCETTA, Optimal Control and Viscosity Solutions of Hamil-

ton - Jacobi - Bellman Equations, Systems & Control: Foundations and Applications,

Birkhauser Verlag (1997).

[4] E.N. BARRON, R. JENSEN, Semicontinuous viscosity solutions for Hamilton - Jacobi

equations with convex Hamiltonians, Comm. PDE 15 (1990).

[5] A. BELLAICHE, The tangent space in sub - Riemannian geometry, in Sub-Riemannian

geometry. Edited by A. Bellaiche and J.-J. Risler. Progress in Mathematics, 144, Birkhauser

Verlag, Basel (1996).

[6] R. W. BROCKETT, Control theory and singular Riemannian geometry, in New directions

in applied mathematics, P. J. Hilton and G. J. Young eds., Springer - Verlag (1982).

[7] I. CAPUZZO DOLCETTA, H. ISHII, Hopf formulas for state dependent Hamilton - Jacobi

equations, to appear.

[8] L.C. EVANS, Partial Differential Equations, Graduate Studies in Mathematics 19, Amer-

ican Mathematical Society, Providence RI (1988).

[9] E. HOPF, Generalized solutions of nonlinear equations of first order, J. Math. Mech. 14

(1965).

[10] S.N. KRUZKHOV, Generalized solution of the Hamilton - Jacobi equations of eikonal type,

Math. USSR Sbornik 27 (1975).

[11] P. - L. LIONS, Generalized solutions of Hamilton - Jacobi equations, Research Notes in

Mathematics 69, Pitman (1982).

[12] J. J. MANFREDI, B. STROFFOLINI, A version of the Hopf - Lax formula in the Heisen-

berg group, to appear in Comm. PDE.

[13] A. NAGEL, E.M. STEIN, S. WEINGER, Balls and metrics defined by vectorfields I: basic

properties, Acta Math. 155 (1985).

[14] L.P. ROTHSCHILD, E.M. STEIN, Hypoelliptic differential operators and nilpotent groups,

Acta Math. 137 247 - 320 (1976).

[15] A. SICONOLFI, Metric aspects of Hamilton - Jacobi equations, to appear in Transactions

of the AMS.

[16] S.R.S. VARADHAN, Large Deviations and Applications, Society for Industrial and Applied

Mathematics, Philadelfia PA (1984).

Conservation of critical groups in a weaker

topology for functionals associated with

quasilinear elliptic equations

Jean-Noel Corvellec and Hamid Douik

Laboratoire MANO, Universite de Perpignan

F-66860 Perpignan cedex, France

and

Lehrstuhl C fur Mathematik, RWTH Aachen

D-52056 Aachen, Germany

Email : corvelle@univ-perp.fr ; douik@mathc.rwth-aachen.de

Abstract

We prove that for a class of continuous functionals on

HQ(CT)

associated to

quasilinear elliptic equations (!) C R", a bounded domain with smooth boundary),

the critical groups at the origin coincide with those of their restriction to

(Cl)

L°°(Q).

1 Introduction

Let H be an open bounded subset of R

with smooth boundary (n > 3), and let /

(fi) -»Rbea functional of the form:

/(«) = I f \Du\

dx - f G{x,u)dx.

2 Jn Jn

In [6, 7], Chang has proved that under some natural conditions on G, so that in

particular / is of class C

, then the critical groups of / at an isolated critical point agree

with those of its restriction to

CQ(0).

Such a result can be used to study the existence

of weak solutions of the semilinear elliptic problem associated with /, by combining the

sub-

and super-solution method with the variational one; see also [3, 1] for the particular

case of local minima. We intend to yield a similar result concerning the critical groups at

the origin for functionals / :

(Q)

—>

R of the form

f(u) = - / \^ aij(x,u)DiuDjudx

—

/ G{x,u)dx,

352

353

associated with quasilinear elliptic equations. We note that if, under natural assumptions

on the dij's and G, such a functional is continuous, it fails to be differentiable, or even

locally Lipschitzian, see [4, 5]. However, we may use the methods of the nonsmooth

critical point theory initiated in [14, 11, 8], in order to follow the lines of proof of [6],

combining the "regularity" of appropriate deformations of a neighborhood of the origin

with an appropriate version of the Morse splitting lemma.

In this paper, we provide a first step in our study (to be further developed in [12])

by establishing that the critical groups of our functional at the origin coincide in the

topologies of

HQ(Q,)

and H^itl) fl L°°(Q). Our main result is in section 4, the proof of

which is based on the Morse lemma that has been recently established by Lancelotti [17],

and that we recall in section 3, and on the existence of appropriate "regular deformations",

which are constructed in section 2 in a fairly general setting.

2 Regular deformations

In this section, we recall from [5] some features concerning a general class of functionals of

the Calculus of variations, and we prove a deformation result for this class. The technique

of proof follows the lines of that of the abstract deformation theorems [11, Theorem (2.8)]

and [9, Theorem 2], dealing with arbitrary continuous functionals on complete metric

spaces, the novelty here being that the deformation obtained is regular, by which we

mean assertion (d) of Theorem 2.3 below.

Let fibea bounded open subset of R

(n > 3), and let / :

H&{Q)

-> R be defined by:

f(u) := / L(x,u, Vu)dx,

where the function L : !] x 1 x R° -> R is C^-Caratheodory, i.e. x i—> L(x,s,^) is

measurable for every (s, £) e R x R

, (s, £) i-+ L(x, s, £), is of class C

for a.e x e ft, and

satisfies the following growth condition : |L(a:, s,£)| < a

(x) + foo(I

:r5

l£|

f°

r a

x 6 H and every (s,£) £ 1 x I", where ao 6 L

(i7) and b

> 0. Under these conditions,

the functional / is well-defined and continuous on

HQ(Q).

Assume further that there

exists ai G Ll

(fl) and &i 6

L™

(Q)

such that

|D.L(a,a,OI < «i(*) + h(x)(\s\^ + |£|2),

(

L(x,s,0\ < a^ + MxXIsl^ + KI

Then, for every u e

HQ(Q.)

we have that D

L(x,u,Vu) e

{£l),

V^L{x,u,

Vu) G

LlJP^W

so that : -div(V

L(x,w, V«)) + D

L{x,u,Vu) €

V'{Q).

Definition 2.1 We say that u is a weak solution of

, . J -div(Vi;L{x,u,S7u)) + D

L{x,u,Vu) = 0 in ft

\« = 0 on dQ,

354

ifue H^(n) and -div (V

(

L(x,u,

VM))

+ D

L(x,u, Vu) = 0, in V(U). For u € H&(Sl)

and w e H£(Q) n L°°{Q), we have:

f'(u)(w)

:= Mm

A!i±HjliM

f [V

€

L(x,u,Vu)-Vw + D„L(x,u,Vu)w]dx eR.

Clearly, w

H->

f'(u)(w)

is linear, while u >-> }'{u)(w) is continuous, as follows from

(1).

In particular, we see that u e HQ(Q) is a weak solution of'(*) if and only if f'(u) = 0

in tte distribution sense.

Let now V and W be (closed) linear subspaces of

HQ(9.)

such that V C L°°(fi)

and fl^fi) = V © W (so that, also, ^(fi) n L°°(n) = fe(lfn £°°(fi))), and let

Pv : i?o(n) —> V be the projection onto V. We denote by ||«||i,2 := ||Vu||2 the norm

HQ(Q),

where ||-||

is the norm of L?(Q), and by ||u|| := |H|

+ ||u|l<» the norm in

#o(n) 0 L°°(Q.), where H-]^ is the norm of L°°(fi). The open ball, closed ball, and

sphere in i?Q(f2), centered at u e

i?Q

(f2) with radius 8 > 0 will respectively de denoted

by B(u; 8), B(u; 8), and dB(u; 8). li X C H£(Q),

will be the closure of X, and we let

:= XnL°°(Q), which we shall consider as endowed with the

topology

ofHQ(Q)<~)L°°(Q).

Set:

f\{u) := su

{f'(u)(w) :weWn L°°(fi), ||u/||i.2 < 1}.

According to the above mentioned properties of

f'(u)(w),

we have:

Proposition 2.2 Let u e

HQ(0,).

Then, \d

f\(u) € [0,+oo], and if \d

f\{u) > a > 0,

there exist w G W

L°°(Q,)

with ||w||i,2 = 1 and 8 > 0 such that

\fv e B{u;8),Vt 6

[0,8]

: f(v + tw) < f(v)-at.

Given X C H^tt) and a continuous function c : X

—>

K, we set:

:= {u e X : /(«) <

c(u)}.

Theorem 2.3 Let H^(Q) = V ® W with V C L°°(Q) and f : H^(Q) -> R as aiow.

Assume £/ia£ /(0) = 0, and that there exist r, p > 0, such that:

VveVn 5(0; r),

VUJ

e W n dB(0; p) : /(v + w) > 0. (2)

Set X := [(F nB(0;r)) + ((ffl 5(0; p))\

/°, and fe£ further a : X ->

]

- oo,

6e a

continuous function with a(0) = 0,

SMC/I

iftai:

Vu, v! E X

pv(u) = Pv(w') =>• a(u) = a(u'), (3)

inf |aV/| > 0, for every

neighborhood

U of f

. (4)

Then,

for Y := X or Y := X \

{0},

and for any open neighborhood U of f DY in Y,

there exists a continuous map n

Y x [0,1]

—>

Y such that

(a) 7?(u,0) =u;

(b)

uef

=> n(u,t)=u;

(c)

n{u,l)eU;

(d) 77(^00 x [0,1]) C Y

, and n :

Yoo

x [0,1]

—>

Y^ is continuous.

355

Proof.

We first consider the case of Y = X. Let U be an open neighborhood of

/°;

we may of course assume that X \ U ^ 0. Let g : X \U

—>]0,

+oo[ be defined by

g(u) := f(u)

—

a(u), and : 7 := inf g, 0 < n < inix\u

\dwf\-

Let u £ X \ U be such

that <7(u) > 7. According to Proposition 2.2 and (2), we find w e Hg(Q)

L°°(Q) with

||w||i,2 = 1 and S > 0 such that:

Vt€[0,6]:u + tweX\U and f(u + tw) < f{u) - fit.

Letting

:=limsu

^-y)

denote the strong slope of g at u [13], we thus have, taking (3) into account:

i„ ,, N^ ,. q(u)

—

q(u + tw) ,. f(u)

—

f(u + tw)

\Vg\{u) > limsup =^—— - = limsup

-^-t—^ ^ > /j,

It follows that </

-1

(7) 7^ 0, according to Ekeland's variational principle [16]: see [2,

Section 2] for more details. We conclude that 7 > 0; thus, considering 0 <

< 7, and

setting ai(u) := a(u) + 7^ a(u) := a(u) + 7 for u 6 X, we obtain:

6 X, f(u) < a(u)

==>ueU.

(5)

Let now a > 0 be such that \dwf\{u) > c, for every »6Z:=I\ /

. For ueZ, let

e W n L°°(n) with |K||i,

= 1 and S

> 0 with B(«;

2<5„)

C #<}(fi) \ /

such that

Vv € B(u; S

),\ft e

[0,

]

: f{v + tw

) < f(v) - at,

according to Proposition 2.2, so that

v«

B(u;

<$„)

n x,

[o,

<y:

v + tw

e z.

Consider a locally finite refinement {Vj

\ : j G N,

e A^} of the covering {B(u; <5„/2)n

X : 1/ £ Z} of Z, with the property that V^x n

jtfi

= 0 for every j e N and

/i e Aj,

X ^

(J-

Let {^

A : j € N,

£ A.,-} be a partition of unity subordinated to the covering

\}.

For every

(j,X),

if V^, C B{u

x,&u

j2), we set

to,-,*

:= u>

Uj>

and

•=

UjX

Since u

t—>

min{5j^

: u

^J,A}

positive and lower semicontinuous on Z, we find a

continuous function T\ : Z —>]0,+oo[ such that 2r

(u) < minffij^ : u G

FJ,A}

for every

€ Z.

Define recursively a sequence of maps 7^ : {(u,t)

Z, t G [0,Ti(u)]}

—»

Z by:

t,A(u)tlUi,A if U G

VI,A

if U^UASAIVU,

356

ft > 2. If u € Vh,u for some ft > 2 and ^ 6 A;,, then

HW^M - u||

li2

£ £ M«)

* <

n(«)

,»-

This shows that Hh-i(u,t) € B(u

htll

htli

), whence Hh is well-defined and

f(H

(u,

t))

< /(«) - * (£ £

))

*•

Since the family {Vj^} is locally finite, for every u € Z there exist an open neighbor-

hood

and h

e N, such that Tih(v, t) = Hh

(v, t) for v 6 A/' and ft > ft

. We can

thus define r}

: Z x

[0,

+oo[—> Z by

»h(u,t) ~ lim W

(i;,mm{t,r

(M)}).

For v e M, we have ||»7i(i>,t) - *>||i,2 < min{£,

Ti(V)},

while if u 6 Z^,, then Th(/M) €

Zoo for i; € J\fr\L°°(fl), and

|l»7

(i'

— v\\ < min{t, Ti(v)} maxi<j<

feu

\\iUj,Xj II

for such v and

for some (w^ )!<,</,„ depending only on u. This shows that rj

as well as its restriction

from Zoo x [0,+oo[ to Z^, are continuous in the respective topologies. Moreover, for

every (u,t) € Z x [0,Ti(u)], we have py(r;

(M,t)) = ?ty(u), and f(f]i(u,t)) < f[u)

—

at.

Now, define recursively rj

: Z x

[0,

+oo[—• Z and r

: Z

—>]0,

+oo[, ft > 2, by

„ /„ rt _ / %~i(

>*) if 0 < i < r

_! (u)

Th(u) = r

_i(M) + nirj^u, r

_i(u))).

Clearly, each r/, is continuous, and so are the r)

's and their restrictions from Z^ x

[0,

+oo[ to Zoo. For each u £ Z and each ft € N, we have

Pv(Vh(

*))

?v(

) f°

every

t > 0, and:

0 < i < T

(u) =• /fa

(u,

<))

< /(u) - at, (6)

ll»//i+l(Wi1"h+l(w))

~ Vhi

,Th{u))\\l,2 < Tl{ri

(u,T

{u))) = Tfc+^U) -

Tfc(u).

(7)

We infer from (6) that

T/,(U)

< (f(u)

—

a(u))/a for each ft € N, so that there exists

r(u) := lim^ r

(u) € ]0, {f(u)~a(u))/cr], and it is readily seen that r is lower semicontinous

on F. We then obtain from (7) that (%(«,

T^(U)))

C (p^(u) + TV) n Z is a Cauchy

sequence. Hence, letting u := lim

(u, Th{u)), we see that f(u)

•=•

«i(u) = Qi(u) — for,

otherwise, we have u € Z while Ti(u) = 0, which is not true. Consequently, we can define

Z -> N

{0} by:

e{u)

( ° if /(«) < "(w)

'' \ min{ft 6 N :/(?7

(u,T/,(u))) < a(w)} otherwise

and clearly, 6 is upper semicontinuous with 6{u) < T(U) for every u S Z. Let then

/? : Z ~+]0, +oo[ be continuous with 6 <

< T, and define

: Z x [0,1]

—>

Z by:

ij{u,t) := lim r)

(u,t/3(u)).

h—»oo

357

For each u e Z, there exists h

e N such that r

(u) > (3{u), hence there exist an

open neighborhood JV of u such that Th

{v) > P{v) > t(3(v) for all v e Af, so that

f)(v,t) = rj

^(v,tp(v)) for such u. This shows that

is well defined, continuous, and so

is its restriction from Z^ x [0,1] to Z^. Finally, let 7x < 7

< 7, set a

(u) := a(u) + 7

(u e X), and let 1? : X

—•

[0,1] continuous, such that $(u) = 0 if f(u) < a

(u), i?(u) = 1

if f{u) > a(u). Define r]: X x [0,1]-> X by:

•q{u,t)

:= J

7?(u, tf(u)) if u e Z

•u

otherwise.

Then,

has the required properties: we just verify that (d) holds. Indeed, if f(u) <

a(u),

then f(rj(u, 1)) < f(u) < a(u) = a(r](u, 1)), so that ?7(«, 1) G [/, according to (5).

If f{

) >

(

)i then i9(u) = 1, it follows that

ri{u,l) =

T!(U,

1) = lim ij

(u,f3(u)) = r)

(u,0(u))

for some h

> 0(u) (since 6(u) <

j3{u)

T(U)),

and

f(r,(u, 1)) = /(77k>,/?(«))) < /(»/,(„,(«,»(«))) < a(u) =

0(77(1*,

1)),

so that r)(u, 1) e [/ again. We now consider the case when Y := X \

{0}.

We first observe

that since a(0) = 0 and (3) holds, we have:

ueX\r=^p

(u)^Q. (8)

Let U be an open neighborhood of f \

{0}

in X \

{0}.

For deN large enough so that

B/,

:= B(0; l/h) n /° C X, say h > h

, we have that f/ U B

is an open neighborhood of

f in X. As in the beginning of the

proof,

we thus find a nonincreasing sequence (7/,)h>/i

of positive real numbers such that

u£X\(UuB

)=*f(u)>a(u)+j

, (9)

and we may assume (without loss of generality, indeed) that 7

—•

0. Define a

pv(X) \

{0}^]0,+co[ by

.= /7fc. iff^B(0;l/M

' \TM-I if « e B (0; l/h) \ B (0; l/h + 1), h>h

Since a is lower semicontinuous, we find a continuous function a : Pv{X) \ {0} —>

]0,

+oo[ such that a < 5, and we define a continuous function a

—>

R by:

^„\

._ /

(

) +

(Pv (")) if Pv (u)

-\0 ifpv(«) = 0.

Then, a(u) = a(u') whenever pv(u) =

(u'),

and f(u) > a(u) > a(u) for u €

\ U, as follows from (8), the fact that ||u||i

> ||pv(«)||i,2, (9), and the definition

of a. Considering ai : Y —> R continuous such that a(u) < «i(u) < a(u) whenever

358

Pv(

) 7^ 0 and ai(u) = ai(u') whenever py(tt) = py(u'), we find, according to (4),

another nonincreasing sequence (cr

)

of positive reals converging to 0, such that

inf \d

f\ > a

Y\(f°lUB

)

and we proceed in a similar fashion as for the definition of a, in order to define a continuous

—»

[0,

+oo[ such that a{u) = cr(u') whenever py(u) = pv(u'), and

f\(u) > a{u) > 0, for every u e Y \ f

Then, we can proceed in a similar way as before (replacing a with

<J{U))

in order to

construct the required deformation

r).

•

3 A generalized Morse splitting lemma

In this section, we consider a functional on Hg(Q) of a more specific form than that

considered in the previous section, and for which we recall a generalized Morse lemma

from [17].

Let fl be a bounded open subset of

R™

(n > 3) with d£l of class C

. For every

1 < i, j < n, let dij : S] x R -» R be such that:

(a.l) dij is of class C

, and the function s i-+ a.ij(x,s) is of class C

for a.e. x €

Q.;

(a.2)

ctij

= aji for every 1 < i,j < n;

(a.3) there exists v > 0 such that for a.e. x € fi, for all s £ R, and for all £ 6

R™

n n

^a„(x,s)^

> v^£

;

(a.4) there exists C > 0 such that for a.e. a; 6 fi, for all s e R and all i,j, k

\a-ij{x,s)\

<C, \D

i:j

(x,s)\ < C, \D s)\ < C, \D

aij{x,s)\<C;

(a.5) for a.e. x €

SI,

for all s € R, and for all £ € R"

Let also p : Q x R

—>

R be such that

(g.l) for every s e R, x

i—>

j(s, s) is mesurable; for a.e. x e O, s i-> p(x, s) is of class C

;

<7(z,0)=0;

(g.2) there exist b e R and 0 < p < ^ such that for a.e. x e ii and for all s e R,

|£

</(MI<6(l +

|s|

359

Let / : HQ(Q)

—>

K be the continuous functional defined by:

f(u):=-

I y_]

ij(,

)DiuDjudx

—

I G(x,u)dx, (10)

where G(x, s) :=

f*g(x,

t)dt. Observe that u 6 HQ(Q) is a weak solution of the problem

(*) associated with / if

71 1

—

\] Dj{aij{x,u)Diu) + - 2_\ D

j(x,u)DiuDjU = g(x,u) in P'(fi),

and that since g(x,Q) = 0, this problem has the trivial solution 0 €

HQ(£1).

Let Q :

fl^fi) -• R be defined by

Q(u) := / y^ a

j(x,0)D

uDjudx

—

/ D

g(x,0)u

dx.

ij=1

(11)

Due to (a.4) and (g.2), Q is a well defined continuous quadratic form. We can write

Q(u) — (Au,u), where

Au :=

—

2_, Dj(a

j(x,0)D

—

g(x,Q)u,

and, taking also (a.3) into account, the elliptic operator (with O-Dirichlet boundary con-

dition) A possesses a nondecreasing sequence (A^) of eigenvalues (repeated according to

multiplicity) diverging to +oo. Then,

m(f; 0) := min{j e N : Xj > 0} - 1

is equal to the strict Morse index of Q, namely, the supremum of the dimensions of the

subspaces of

HQ(Q)

where Q is negative definite, and

m*(/; 0) := min{j 6 N : Xj > 0} - 1

is equal to the large Morse index of Q, namely, the infimum of the codimensions of

the subspaces of HQ(Q) where Q is positive definite. We thus have the decomposition

tfi(fi) = V 0 W, where V C ff

(fi)

(n), dimF = m*(/;0) < oo, and

£V,

\tw€W:

/ V^ a,ij(x,0)DivDjWdx

—

/ D

g(x,0)vwdx = / vwdx = 0,

Jn ;T"i J n •> n

3//

> 0, Viu G W : Q(w) > n\\w\\

Under the above assumptions on the o^'s and on g, the following generalized form of

the Morse splitting lemma is proved in [17, Theorem 6.17, Lemma 6.15]: