- Email: [email protected]
On existence and uniqueness of solutions of Hamilton-Jacobi equations
ON EXISTENCE
AND UNIQUENESS OF SOLUTIONS JACOBI EQUATIONS
OF HAMILTON-
MICHAEL G. CRANDALL klathemarics Research Center. University of Wisconsin-Madison. !vLIadison.Wisconsin 53706. U.S..1
and PIERRE-LOUIS LIONS Universite de Paris-IX-Dauphine.
place de Lattre-de-Tassigny.
75775 Paris Cedex 16. France
(Receioed in revised form 20 June 1984; received for publicarion j June 1985) Key words and phrases: Hamilton-Jacobi equations, first order nonlinear partial differential equations. existence, comparison theorems, viscosity solutions. moduli of continuity.
ISTRODUCTION OCR MAIN purpose here is to establish some uniqueness and existence theorems for HamiltonJacobi equations. We will focus on the Cauchy problem
U, + H(x, f, U, Du) = 0 in Rv X (0, T). u(x, 0) = f&),
(CP)
and the stationary problem u + H(x, ~1,Du) = 0
in R.v.
(SP)
in which H is a real-valued function of its arguments, x denotes points in 5&l’,Du stands for the spatial gradient (llX1, . . . , u,,~) of 11,which is itself a real-valued function of either (x, t) or x as appropriate. The uniqueness follows from comparison theorems of maximum principle type. The comparison theorems we present here correspond to some results proved in special cases by Ishii in [9] and used by him without proof in more general cases in [8]. These results concern the comparison of solutions which are uniformly continuous in x-but perhaps unbounded-in cases where the Hamiltonian H is restricted by appropriate continuity conditions, but not otherwise by growth conditions. We will give complete proofs in a natural generality which strictly includes the corresponding statements of [8]. Moreover, it would seem to us worthwhile to present our complete proofs even in the special cases used in [9]. For other sorts of uniqueness results for unbounded solutions see Ishii [9] and Crandall and Lions [a, 51. As regards existence, we will prove existence results in the same new generality for which we establish the uniqueness. The main tool used for this is estimated on the modulus of continuity of solutions of (CP) and (SP). These estimates are obtained by using the comparison results and modifications of the elegant ideas of Ishii [8]. The notion of “solution” of the differential equations in (CP) and (SP) that we will employ is that of a “viscosity solution”. We will not explain this notion here and refer the reader instead to Crandall, Evans and Lions [2] for easy access to definitions and some proofs of the 353
sort presented here in the simplest situations and to Crandall and Lions [3]. where v-arious equivalent notions of viscosity solutions were presented and the first uniqueness proofs were given. The question of existence of solutions of Hamilton-Jacobi equations has a long history,. We refer the reader to the book Lions [lo] for references to the substantial ~.ork which predates the notion of viscosity solutions. Existence theorems in the general spirit of this paper were established in model cases in Crandall and Lions [;I. Considerable generality, was achieved in the results of Lions [lo. 111 (’m which boundary problems were considered-vve do not discuss existence under boundary conditions here, although it is obvious that one could). some new results (in 2-v) were obtained bv Souganidis [13] and then arguments were introduced in Barles [l] which brought the existe-nce theory to a new level of generality. Ishii [S] (using estimates of moduli) and Crandall and Lions [4] (relying partly on Barles’ methods) were the first works concerning the existence of (possibly) unbounded viscosity solutions. The rev,ieu paper [7] outlines other aspects of the theory of viscosity solutions and contains a moderateI>, current bibliography. The test begins with a Section 1 which is devoted to preliminaries and statements of the main existence and comparison theorems. The comparison results are then proved in Section 2. Section 3 is devoted to studying the moduli of continuity of solutions of (CP) and (SP). The results on the moduli of continuity quickly imply the desired existence theorems. 1. PRELI.LIISARIES.
NOTATION
AND
STATEXLENTS
OF RESCLTS
We will use the following notational conventions throughout. The set of continuous functions which map a metric space 0 into the reals will be denoted by C(Q) and the subspace of C(Q) consisting of uniformly continuous functions will be denoted by UC(Q). Functions of modulus of continuity type will be denoted by the letter m as n.ell as m nith various subscripts (e.g. m R. mH. etc.). Such functions m map [0, -/-) into [O. x) and satisfy: (i) (ii) (iii) (iv)
m is continuous and nondecreasing; m(O+) = 0: m(n + b) S m(a) + m(b) for 0. b 2 0; m(r) S m(l) (r f 1) for r 3 0;
(1.1)
where (iv) is in fact a consequence of (i) and (iii). We will call any such function “a modulus”. When Q is a subset of F?.‘, UC,(Q x [0, T]) denotes the space of those 11E C(Q X [O. T]) for which there is a modulus m and an r > 0 such that lU(X, t) - u(y. t)l G m(]x - y]) forx, Finally,
in what
follows,
y E R, ].r - y/ s r and f E [0, T].
T > 0 and BR(z) denotes
the closed R-ball
centered
at 2 E 62”.
will typically
be restricted
BR(Z) = {X E 5Qv: /x - z] =z R}, while intB,(z) the following
is its interior. The Hamiltonians set of conditions:
(HO) HE C(lQ’ x [0, T] x II2 x @j; (Hl) for each R > 0 there is a modulus x
we will consider
mR such that for all (x, t,~),
BR@)
jH(x. f. r, p) - H(y, f, r, q)l 5 m,(b
- q! + 1~ - :I);
by
(y, f, 4) E 2’ X [O, rl
On tx~stence
and uniqueness
oi solutions
of HamIlton-Jacob1
equations
35
(HZ) H(x. r. r. p) is nondecreasing in r for all (x. r. p) in 8.v X [0, T] x A’: (H3) there is a modulus mH such that for all (t, r) E [O. T] x 72 and all X. )j E 2v and E.2 0 H(s. f. r. E.(x - y)) - H(y. I. r. 1.(x - y)) 3 -m,(+ Remark 1. The assumption
(H3) is a replacement
- 1’1’ + ix - !I).
for the stronger requirement:
(H3)’ there is a modulus mH such that for all (t, r.p) E [O. T] x A x 2’ and all x. .” E 2’ lH(x, f, r-p) - H(y. f, r,p)i c mH(lx - .4(1 + lpi)): which was introduced in [3] and has been used since then. The significant observation that the original proofs of uniqueness go through under (H3) (although one should choose q to be radial in, for example, the proof of [3. theorem 11.11, as is done later in [3]) is due to R. Jensen who was led to this remark by considering the uniqueness question for Hamilton-Jacobi equations in infinite dimensions in an ongoing investigation. It is easy to see that (HO)-(H3) is strictly weaker than (HO), (Hl), (HZ), (H3)‘. For example. if N = 1, H(x. r.~) = b(x)p satisfies the weaker set of conditions if and only if b is bounded and uniformly continuous and b(x) + cx is nondecreasing for some c, while the stronger conditions replace this last requirement with Lipschitz continuity of b. The analogous remark holds for general N. (Later vve will comment on how (H3) itself may be significantly relaxed.) Finally, the requirement that mH be a modulus may seem restrictive, but it is not-if this estimate holds with any nondecreasing function tending to 0 at O+, then it holds with a modulus. The uniqueness and existence results for (CP) are: THEOREM 1.
Let H satisfy (HO)-(H3).
Then:
Comparison. Let R be an open subset of iw”, 0 its closure and aS2 its boundary. o E UC,r(5? X [0, 7’1) satisfy the differential inequalities
Let U.
Il, + H(x, f, u, DU) G 0 and 0 < u, + H(x, f, u, Du) on R X (0, T) in the viscosity sense. Assume also that u(x, 0) 6 u(x, 0) on R and u(x, t) s u(x, t) on aR x [0, T].
(1.2)
Then 14c u on R x [0, T]. Existence. Let ~j E UC(IB”). Then (CP) has a solution u E UC,r(W*vx [0, r]) in the viscosity
sense. Remark 2. Theorem 1 extends, via the substitution u = e -“u, to Hamiltonians H(x, r, u,p) + Arc satisfies the assumptions in place of H for some A.
H for which
Of course, the “comparison” theorem when formulated in this way gives uniqueness and much more-it also implies continuous dependence and will be used to estimate moduli of continuity, etc. The parallel theorem for (SP) results upon interpreting (HO)-(H3) for a
356
>I.G. CR.GDAL.Lend P.-L.LiO\j
function H: X,v x R x R.v+ 2, bv regarding considered above. We have: 2. Let H E C(2’
THEOREM
Comparison. inequalities
Let
Has a (r-independent)
special case of the functions
x 2 x Zv) satisfy. (HO)-( H3). Then:
Q be an open
subset
of 2’
and
M. u E UC’(o)
satisfy
the differential
u f H(.r. II. Du) < 0 and 0 G u + H(.r. ~1.Du) on 0 is the viscosity Exisrence.
sense and II 4 u on 3Q. Then
(SP) has a solution
(1.3)
II s u on 0,
II E L’C(;;iL).
Remark 3. Assume that dR is of class C2 and v, denotes the inuard normal to R at .YE 3Q. Then the assumed comparison u G u of aQ x [O. T] in theorem 1 and on 6Q in theorem 2 can be weakened by replacing dR by the set of .r E aQ for which A - H(.u. f. II. p L Av,) is not nonincreasing for some (f. ~1.p) E [O. T] x 2 x 5%”and \,ariants of theorems 1 and 7 remain true. One needs to strengthen (H3) near the boundary a bit-see Crandall and Nexvcomb [6]. The next lemma collects some simple estimates used throughout the test for con\,enient reference. It may be skipped at this time. LE~IXIA 1. Let
~FZ
be a modulus.
Then:
(i) Let E, 6, r > 0. m(1) > 0 and c’,+ nz(r) 2 r’/‘s. Then. r G 7(r~z(l)~)’ (ii) If E 2 0, F>
’ andr’;E
nz(1) and k’(;f) is defined
s a f
Proof.
rF1(?(VZ(1)&)’
‘).
small. (1.4)
for ;/ > 0 by
K(;j) = sup{vr( yEr -I- (r/F)’ then K(O+)
if E and b are sufficiently
1’) - r: 0 s r s F}
= l,ifn K(;/) = 0.
We first sketch
the proof
r2 S e(m(l)(l
of (i). The assumptions
and (l.l)(iv)
+ I) f S) =5 4E mas( d. m(l).
yield
m( 1)r)
and. choosing the various possibilities for the mas on the right. the Lvorst estimate on r uhich arises for small 8, E is the one claimed in (1.1). Then the second estimate arises from the monotonicity of m. (Obviously these estimates can be improved. but \ve \\ill not need to do so.) We just sketch the simple proof of the (aivkward looking) assertion (ii). Let I’.. be the maximum point of f:,(r)
=
TFZ(
;.fEr f (r/F)’
:‘) - r
2 m(O) - 0 = 0. If (r:./F)’ z’has a limit point 8 as ;’ ; 0 and 0 < 8. then on [0, F]. Clearlyfy(r,) the corresponding limit superior of f:,(r,) is at most nr( 1) - F < 0 (so this cannot be). ivhile if B = 0 then the limit superior is at most m(O) - 0 = 0. Since K(;./) =f.,(r..). u.e are done.
We conclude this section with a useful lemma which illustrates the advantage one can take of the linear dependence on the time derivative in (CP). This is a convenient packaging of a standard argument. (For a sketch of a proof in a special case see Ishii [8, lemma 3.31.) LEMXA 2. Let R be an 0 E C(Q x (0. 7)) satisfy
open
of w‘“, H, G E C(Q x (0, T) x R x RL)
subset
and
U.
LI, + H(x, t. II, Du) s 0 and 0 G u, + G(x. f. 0, Dv) on R x (0, r). Define
2(x. y. r) on R
x R
(0. T) by z(x, y. r) = U(X, f) - u(y. 0. Then
x
Zl + H(x, f. U(X, f), D,z) in the viscosity
- G(y, t. o(y, t), -D,z)
c 0 on Q X Q X (0, T)
sense.
Proof. It will suffice to show that if q E C’(Q X R x (0, T)) and (x7 y, f) has a strict local maximum qi(“.
4x,
at (a, b, c) E R
t) - u(y, t) - e(x, .l’, t) x R
x
(1.5)
(0, T), then
b, c) + H(a, c, ~(a, c), D,rq(a, b, c)) - G(b, c. u(b. c). -D,q(n,
b, c)) s 0.
(1.6)
Put d(,u, y, t. s) = max(]x - aI, ly - 61, /r- c . IS - cl). Since the maximum such that
is strict,
there is an r. > 0 and HE
U(X. t) - u(y. f) - q(x.y, for 0 < d(x, y. f, f) G ro. We claim: given r > 0 there
qJ(,y, y, t, s) =
has a local maximum
point
E >
- H(d(x,y,t.
t))
(1.8)
0 such that
u(x, t) - u(y, s) -
rp(xzL’? 0 - (t -
d2b
(1.9)
(x,, yr, t,, s,) which satisfies +,,Y,,
Assuming
C([O. r,]) with H(r) > 0 for 0 < r < r.
t) s rc(a,c) - o(b, c) - ~(a. b,c)
is an
(1.7)
the claim is correct
for the moment
t,, J,) 6 r.
(1.10)
we then have.
by the assumptions.
If these inequalities are added, r is sent to 0 and (1.10) is taken into account, It remains to produce (xr, yI, t,, 3,). Fix r C r. and choose E so that ~U(y. f) - U(y, S)i + lU(x, f) - U(X,S)] + ]cp(x,y, t) - V(x,y,
s)i == (t -
s)‘/E
the result follows.
+
H(r)/2.
(1.11)
58
41. G. CR.ANDALL and P.-L. LIOXS
for n(x. y. r. s) c rO. This is clearly possible. Let Y be given by (1.9) and d(x. y. f. s) = r. Then either d(x. _v. t, t) = r or d(.u. ,v, s, s) = r. Assume d(x. y. I, t) = r (the other case is treated a similar way). Then using (1.8) and (1.11) we have V(X,_y. 1, s) 5 U(X, f) - u(,V. t) - q(X,)‘.
f) - (u(,V. 5) - u(_V. f) t (f - 1): !‘E)
s Y(a. 6. c. c) - H(r) - ;oi,V. s) - U(_V,f)/ - (f - $)2;‘& =z Y(a. b. c, c) - H(r)/2. so the maximum of Y over the set (f c r must occur at an interior maximum satisfying (1.10). 7. PROOFS
OF THE CO.LIPARISOX
We will first prove
essential
theorem 1. which is more difficult step in the proof of the comparison assertion
LEMMA 3. Let the assumptions of the comparison bounded from above on R x [0, T]. Proof
of lemma
3. Since
point,
which is then a local
RESULTS
than theorem 2. The next lemma of theorem 1.
assertion
U, u E rlC,,(Q x [0, ir]), there
of theorem
is a modulus
1 hold. Then
is an
u - ~1is
m and an r > 0 such
that lu(.r, t) - ~(y, t)l + /u(x, r) - u(,v, f): G m(lx - J/) (2.1)
for Jc, y E 0 and 1.t.- y] s r. Let n(,u) = distance(x, Since 11< u on aR x [0, T] and (2.1) holds.
(2.2)
JQ).
we have
U(X, t) - u(y, f) G m(lx - yl) + m(min(d(x),
d(y)) f0r.u. ,VE a and
t E [0, TJ such that ]x - yj 4 r and min(d(x),
d(y))
(2.3)
Thus to bound u - u it will suffice to bound end, choose such an a and set
6 r.
~(11. t) - ~(a, t) for n E Q and d(a) 2 2r. TO this
K = 1 + 2m(r)/r’,
Y(x, y, r) = u(x, t) - u(y, l) - K(jx - al? + jy - al’), 44
= supW(x,
y, 4: (x, y> E &((a,
(2.4)
a>)),
for 0 G t < T. Here &((a,
u)) = {(X, y) E W X W: jx - u/l + ]y - a/* < r’} c G x !3
by d(a) 3 r. In order to prove the lemma, it suffices to show that w is bounded from above uniformly in a. Clearly w is continuous. The main part of the proof is to show that w’ is bounded above on the open set {w > O} = {t E [O, 7-l : w(t) > 0)
On existence
and uniqueness
o!’ solutions
of Hamllron-Jacob
equation>
by a constant in the viscosity sense (and hence in the distribution sense-see I. 111). Since u c u on R x (0). vve have
359
[2. proposition
w(0) 6 sup{u(x. 0) - u(1’. 0) - K(Jx - 1111+ I?’ - ai’) : (x. J) E B,( (a. a))} s ‘m(r). If also w’ G C on {w > 0). we conclude that w(l) s 2m(r) + CT on [O. T]. whence the result. It remains to show that w’ is bounded above on {w > O}. The constant K in (2.1) is chosen so as to guarantee that Y(x,_~.r) < Y(a, a, t) if (x._v) E JB,((a. a)). so there is a point (,f. ,V)E int B,((a, a)) such that w(r) = u(l, f) - u(,:. f) - K(1.f - 1211+ ~!:- alZ).
(2.5)
Of course, f-9 depend on t and u. Let I be an open interval in {IV.> 0). p E C’(I) and w(f) - p(f) have a local maximum at TV I. Let (2.9) E intB,((cr. 0)) be such that (2.5) holds with f = i. Then for all (x, y. f) near (_c.j. r) U(X, r) - u(y, f) - K(jx - 0/Z - ly - a]‘) - p(r) 6 u(X. ?) - u(_V,S) - K(1.f - ui2 - I? - 01’) - p(S). It then follows from theorem
1. lemma 2 and the definition of viscosity solutions that
p’(i) + H(2. I, ~(2, i), 2K(,r - a)) - H(y. i, u(y. i). - X(y
- u)) s 0.
(2.6)
We will use this with the estimates given next to reach our conclusion. Clearly ZKI,? - a]. 2K]j - a/ G 2Kr.
(2.7)
The string of inequalities written next will be explained immediately H(P, i, u(f, F), 2qx
- cl)) - H(j) I, u(j, i). -2K():
below.
- a))
2 H@, F, U(f, T). 2K(f - li)) - H(_V.i. u(.\-, i). -2K()i - a))
(2.8)
- y/ + 2Kj,f - a] - 2Kly - a])
2 -rn2& 2 -mzK,(2r(l
+ 2K)).
The first inequality is due to the assumption w(f) > 0 (which implies n(X. i) > u($, r) > 0) and (H2). In the next inequality, we use (2.7) and (Hl). The final inequality uses I.\:- ?_Is 2r and (2.7). From (2.6) and (2.8) we now know that, in the viscosity sense, w’
ZKr(2r(l + 2K) on {w > O}.
and the proof of lemma 3 is complete. Remark
4. The reader may observe that only (HO)-(H2) were used in the proof of lemma 3 and (H3) was not required. An analogous remark holds concerning the study of (SP) later.
End ofproof of comparison fur (Cl’). Assume the hypotheses of the theorem, holds and that 116 u fails. We may choose a constant c such that
c > 0 and M = sup{u(x, t) - U(X,r) - C~:XE n and 0 G t s i? > 0.
that (2.1) (2.9)
360
41.G. CR;\\D-\LL
Note that M < x by lemma
P.-L Llo
and
3. Next let E. @ > 0 and set
y’)).
Y(.r,,V. r) = u(x. f) - u(_V, f) - Cf - (!x - >$ ;t. L i;(‘.V :
(3.10)
Clearly S(P) = M - sup{Y(.r,
(2.11)
x, t) :x E G. 0 G f 9 7-}
satisfies = 0
(2.12)
y. I) :(x,y. t)E fi x fl x [O.T] and 1s - >! < r} 2 :CI - d(p).
(7.13)
d(o+) and sup{Y(.r,
In view of (2. l), for (x. J, r) E d x Ci=2 x [O. T] satisfying
I.~ - J, G I’ vv.e have
M + m(i.U - yi) - (1-r - y/‘/E + /3(i.Vi’ + ~J,,‘)) 12.14)
2 11(x, t) - u(x, I) - cr i (U(.Y. f) - 1:(_V.t)) - (.\’ - .“,yE + /3(1.yi’ + Iyi’)) = Y(x. )‘, f)
and from (2.13) and (2.14) it is clear that Y attains a positive masimum with respect to the set {(I, y, t) E h X G X [O. T] : Ix - ~1 =2r} at some point (.f. s. 7) if /3 is small (as we hereafter assume). We now proceed to make some simple and standard estimates inv.olving (.c. !:. F). Usins (2.1-l) and (2.13) we have (2.15)
6(/j) + m(]r\’ - I_$ 2 11 - _$/E i /4( .f 1 f i\:l’). From
(2.15)
provided
and lemma
1 we deduce
that (taking
m(l)
+ 0 \vithout
(i)
I.?- ,PI G Z(m(l)E)’
(ii)
~- - ,i2!E =s b(P) + m(2(nr(l)E)1 1.r
(iii)
/?(l,tl’ i !?I’) 6 6(/3) i m(?(??Z(
only that E and ,!3 are small.
loss of generalit>)
‘. (2.16)
?). 1)E)' ?).
Next we check that f> 0. Indeed.
Y(x,,V, 0) = u(.u, 0) - u(x. 0) + u(s, 0) - u(_V. 0) -
((is - .‘.l%
the inequality
t &r
1 - !I:,)
d m(1.V - J’) - (‘.V - viz/E + P(I.yl’ + lvi’)) coupled
with (2.16)
shovvs that Y(_f.,7.0)
so Y(2.L;. 0) 3 M - S(/3) is impossible or d(y) s r. (2.3) yields
G rn(Z(VZ(l)E)’
if E and p are small.
‘) Thus r> 0. Similarly.
if ti(.i-) d r
Y(1. F. r> < 2m(r). nhich cannot be if r is sufficiently small (and we may take it to be so). Lastl!.. (2.16) (i) guarantees that 1-U- ~~1< r if E is sufficiently small. Thus vve can guarantee that (.i-. .\-. F) is an
On existence
interior
point
and umqusnrs,
of solutions
of Hamilton-Jacob1
361
equations
of the set {(x, Y. f) E G x 0 x [O, 7-l : lx - j/ c r}
for vvhich it provided To proceed. put
a maximum
of Y.
p = 2(X - )s)/E, qx = 2p,e, qv = -2py. Since Y has an interior solution yield
maximum
point
at (,?,L;, Q, lemma
(2.17)
2 and the definition
1. u(i. r,>, p + qx) - H()s, I. o(Y, i),p
c + H(i,
+ qJ
We next write a string of inequalities to be used in conjunction them immediately following: We have t‘), p + q.t) - H(j,
H&f. r, 42,
I, u(j, g,p
I, IQ. S),p + q*) - H()?, I, u(Y, i),p
a H(Z
i, u(j,
0, p) - H()?, i, u(g, 0.P) -Pi
The first inequality is due to the monotonicity (H2) inequality arises from (Hl) when R, denotes a bound from (2.16) and (2.17) that
for a suitable
constant
(2.18)
with (2.16) and then explain
+ qJ
(2.19)
- (m,,(lq,l)
!P + /If - r‘j) + mR,(lq,zI)
(d&)jpl
s 0.
+ qr)
2 !f(x,
2 -(m,(!*f
of viscosity
+ m&(lqj
)I>
+ m&?,J)). of H and u(X,?) > o(y,7). The second on ipi, (p + qll, and lp + qJ. It is clear
(2.20)
c C and ]qX1,/qv/ s Cd/p
C as E and /3 tend to 0. Thus for some C we have R, s C/~/E.
(2.21)
The final inequality of (2.19) arises from (H3) and p = 2(Z - Y)/E from (2.17). again we have 11 - y] + 0 and /,r - y/ ip] = 211 -$/E
< 26(P)
+ m(2(m(l)&)r
Using
2).
(2.16)
(2.22)
We are now essentially done: letting /3 J 0 and then E J 0 in (2.19) yields 0 on the right-hand extreme of (2.19) because of (2.20) and (2.22), so (2.18) implies that c G 0, a contradiction. Proof of comparison for (SP). The proof of the comparison common with that for (CP). and where it differs it is simpler. theorem 2 we first prove that: LEMXA 5. Let 11and u be as in the comparison from above. Proof. Let m be a modulus :G)
- 4Y)l +
assertion
result for (SP) has much in Assuming the hypotheses of
of theorem
2. Then u - u is bounded
and r > 0 be such that
14~)-
4Y)l < m(]x - yl) for x, Y E 0 such that IX - y] s r.
(2.23)
362
>I. G. CRUDALL
Analogously
and
P.-L. Lro~s
to (2.3) we have [l(x) - o(y) s m(lx - yl) + m(min(d(x).
d(y)) f0rx.y
such that ~,r - y! G r and min (d(x),
s r.
E B (2.24)
Choose
a point
d(y))
a E 0 such that (7.35)
u(a) - u(a) > m(r), (if there
is no such a we are done) K = 1 + Zm(r),‘r’
and
and then d(a) > r by (2.21). Y(x,y)
The function
Y defined
= U(X) - u(y) - K(1.r - u12 i iv - u;‘).
by (7.26)
satisfies Y(u, a) > Y(x, y) for (x, y) E aB,(u, a) and so it will have an interior maximum (~I?,?_) with respect to this set. We then use the definition of viscosity sub- and supersolutions to find that u(X) - o(j)
+ H(1. u(x), 2Kc.r - u)) - N(y, u(_G). -2K(,?
- u)) s 0.
Since u(i) 2 u(y) (by (2.23) and Y(,f, j) 2 Y(u, u)) we use (Hl)-(H2) theorem 1 to conclude that ~(3) - u(y) is bounded above and then U(U) - u(u) 4 u(i)
now choose
of
- u(V) + 2m(r)
to conclude that u - u is bounded above. To complete the proof of theorem 2, we assume &I = sup(u(x)
as in the proof
that (7.27)
- u(x)) > 0:
E > 0 and /3 > 0 and let (x, I;) be a maximum Y(X, ,v) = U(X) - u(y) - (is - $/&
point
of
+ $0(1X1”+ IylZ))
over the set S = {(x, y) E 0 x G: Ix- y/ =Zr}.which exists since M + m(lx - yi) - (IX -y/‘/E
+ p(iXl’ + iyi’)) 2 Y(X,,V)
(2.28)
on S. Moreover, (2.29)
Y(X, Jq 2 IV2- S(P) where S(O+) = 0. As before, assumptions, if
one
rules
out X, jt E a.S for 6, E, r small
and then,
using
the
p = 2(P - y), qx = 2@, qy = -2py we have &I - c%0 =Z U(X) - u(T) 6 H()?, u(,‘>,p
+ qy) - H(&
u(.f),p
+ qr).
Finally, arguing as in the proof for (CP), lpi is b ounded by C/d& for small E and p, while with E fixed q,c and qv tend to zero like g/p and the proof is completed just as before. Remark 5. One knows, in dealing with bounded solutions, that uniform continuity of u or u (and not necessarily both) suffices to prove comparison results. The analogous statement in
363
On existence and uniqueness of solutions of Hamilton-Jacob1 equations the present of theorem such that
context is that the above proofs easily adapt to prove that the comparison assertions 1 remain correct provided only that u and u are continuous, there is a constant K lU(X, r) - rc(y, f)/ + 10(x, r) - o(y, t)! s K(1 + 1.r- yl)
for (x. t), (y, t) E 0 X [O. T] and one of 11, u lies in UC,,(a x [O. 7-1). The analogous statement for (SP) arises upon lettin g u and u be independent of t. Later we us2 this remark and the existence results of Section 2 to improve the comparison results in the case R = 2.v. 3. LIIODULI OF CONTINUITY XSD EXISTENCE Throughout this section we assume that H satisfies (HO)-(H3) in the case of (CP) and. as before, interpret these conditions and others to be laid below in the case of (SP) in the obvious ways. To begin, following Ishii [8], we label the following structure properties of H: C will be a constant such that ]H(x, t, 0, O)i G C(1 i 1x1)
(3.1)
jH(x, f, U, 0) - H(y, t, U, O)] 6 C(1 + 1.X- _vl)
(3.2)
or for X, y E iRv, K E R, t E [O, T]. The existence of such constants C is guaranteed by the assumptions. Contrary to the previous section, we will begin with the simpler case of (SP) and prove: THEOREM 3. Let HE modulus m depending
C(R” x W x R”) satisfy (HO)-(H3). only on C,
If also (3.2) holds,
then
there
a(R) = m,(R). (where mR is from (Hl)), of (SP) then
and mH (from (H3))
(3.3)
such that if u E UC(R”)
is a viscosity
solution
]u(x) - u(y)1 G m(lx - y]) for Js. y E R’. Moreover,
if (3.1) holds,
is a
(3.4)
then ]I+)]
G C(1 + Ix]) + a(C).
(3.5)
Proof of theorem 3. We begin with the estimate (3.5). The proof, given our comparison result, is essentially the same as the proof of [8, proposition 3.21. The only property of c~ needed is ]H(x, t, u,p) In order
to guarantee
- H(x, t, U, O)i < a(R) for lpi s R.
that u(x) = Alxj + B,
where A, B s 0, solves
u + H(x, U, Du) 2 0, we observe
that
u + H(x, u, Du) 2 Alxl + B + H(x, 0, ADlxl) 2 Ajxl + B + H(x, 0.0) - a(A) 2 Alxl + B - C(l + Ix]) - a(A),
(3.6)
and then take A = C and B = C + a(C). (We are being a little bit formal: how,ever. the subdifferential of ix; at the origin is the unit ball, and so D1.r; stands for an arbitrary v’ector in the ball at x = 0. and the computation is valid.) The estimate (3.5) with II in place of ~II~now follows from comparison. One then estimates u below by -U in a similar vvav to complete the proof. Kext, vve can prove in a similar way that if C is from (3.2). then U(X) - u(y)’ s qu Indeed,
observe
(3.7)
- I‘~ + (C + Za(C)).
that .?(X.!‘) = U(X) - u(y)
is a viscosity
solution
of 7 + H(X, U(X). D,i)
and so the monotonicity
property
z(.~,y)
i
Drz)
U(J). -D,.Z)
(fG,
=
0.
that
- H(y. I.
Ju _ ,xz.v x @v, H(,~,L’,P. 4) = j$
(3.9) and (3.10)
- H(y.
of H guarantees
f H(x, u(y),
DefineH:~~x~.Lx~).~xD~i A
Clearly
(3.8)
-D!z)
s 0 on (2 > 0).
(3.9)
R by 7,~) - NJ.
(3.10)
r, -q)).
impI) -7 + ti(x, y, D.,z, D,.z) G 0 on {z > O}.
and it is also obvious
that fi satisfies
(HO)-(H3)
on %’
(3.11)
X z’.‘. We seek solutions
u(x, y) = A!x - ~‘1’ + B
(3.12)
of u + fi(x,y, D,ru, DLu) 2 0 on Xv,
(3.13)
where A, B 2 0 on 0 < y 4 1. We make some formal (when x = y) estimates below which one can (following Ishii) make rigorous by first replacing js - y!? by (ix - yi2 T E)” ‘) and then letting E-+= 0. We estimate for all r: u + H(.r, r. D,u)
- H(y. r, -D,u)
+ H(x, r, :/AI-u - yiY-‘(x
s Ajs - ).I? + B
- y)] - H(y,
In the case y = 1 we use (3.2) and (3.6) to estimate
(3.14)
r, ;/Alx - yl’-‘(I the right-hand
- y)),
side of (3.14)
below
b>
Alx - yi + B - (C( 1 + /X - yi) + 20(A)) so u satisfies
(3.13)
if A=CandB=Za(C)+C.
Since z < u on a{z > 0}, we use the comparison result for (SP) to conclude that (3.7) holds. We will use the bound (3.7) while estimating the modulus of continuity. Let ;J < 1. We will
On existence and uniqueness of soluuons of Hamilton-Jacobi
equations
compare z and u only on the set {Z >
0)n {IX - ~1< 1)
sowe want z 6 u on the boundarv of this set. Bv (3.7) this will hold if (3.15)
A 3 2c + 2a(C). We guarantee
this by setting, for reasons which will soon be evident. A = max(2C + 2a(C). mH(l) + 1).
(3.16)
Next we use (H3) to estimate the right-hand side of (3.13) below by Alx - yl’ + B - m,z,(yAlx - _v/:’+ jx - _v/)
and, thinking of Ajx - yl’ as r, we see this is nonnegative B 2 K(y) = max{mH( yr + (r/A)“‘)
on {]x - y] < l} provided that - r: 0 G r s A},
Since A > mH(l) by (3.16), lemma 1 yields K(O+) = 0. We now have shown that U(X)-u(y)sK(y)+Alx-y]YforO<;~~land]x-y~
of y and holds. Thus m(r) = inf{K(y) + Ar’: 0 < y =S 1)
provides the desired modulus on [0,11. Proof of existence for (SP). It follows from theorem 3 and the uniqueness that if {H,,} is a sequence of Hamiltonians on w” X R X w” satisfying (HO)-(H3), (3.1) and (3.2) uniformly (with th e same ma, mH, C), &, E uC(Rv) are solutions of 11, +
H&c, u,, Dun) = 0
and H, tends locally uniformly (on R” X W X IRS) to a limit H, then there is a solution u E UC(Rv) of u + H(x, u, Du) = 0 and u,,+ u locally uniformly. Thus existence follows if, given H satisfying the assumptions, we can produce sequences {H,} and {u,}. Let 8 E C?(R) be nonnegative, symmetric, 8(r) = 1 for Irl s 1 and 6(r) = 0 for Irj 2 2. Put H,(s. 1t.p) = H(x, u, 0(lpi/n)p). Clearly H, satisfies the required conditions uniformly and also. for fixed n. H, satisfies (HO)-(H2), (H3)‘. Ishii’s existence theory may be invoked to provide the u,, and we are done. We turn to the Cauchy problem. Modulus
of continuity for (CP) Let HE C(R” x [0, T] X R X ET”) satisfy (HO)-(H3), (3.1), (3.2) and (3.6). Let u E UC,(R” X [0, T]) be a viscosity solution of u, + H(x, t, u, Du) = 0 on W” x (0, T]. Let Co
THEOREM 4.
41. G. CRA>D.ALL and P.-L. Lrow
366
be a constant, lq(x)l
q(x)
= u(x. 0) and m, be a modulus
s CO(l + Ixi) and iv(x) S m,(ix
- y1) fors.
such that
- q(y)1 s Co(l i ;.Y- !:I), and iv(x)
- F(J)
(3.1’)
,v E RV.
Then there is a modulus m depending mH (from (H3)) and T such that
only on C (from
(3.1).
(3.2)).
CU. mF (from
iU(X. t) - “(I’, t)l s m(ix - ,11) for x, !: E 52.’ and f E [O. 7’1.
(3.17)).
(3. IS)
iMoreover, Iu(x, f)I G (1 -I- L)(A&
+ B,,) and lu(s, f) - ~(y, t)i 5 (1 + r)(A,l.r
- Jo f &,).
(3.19)
where A,, = max(C.
C,), B0 = mas(Cli.
C + ?a((1
+ ir)A,,)).
(3.20)
Also for each R > 0 there is a modulus mR depending only on C. Co, ~2,~. u. mH. T and the function F(R) = max{lH(z. r. r. O)l: /=I s R. f E [0, T], /r/ =GA0R + B,,}. such that ~u(x. r) - U(X, s)l d mR(ir - s/) for .K E BR(0) and f, s E [O. ir]. Proof. The strategy of proof is the same as in the previous of (3.19). one shows that
(3.21)
case. To verify the first inequalit).
u = (1 + t) (Ai,ri + B) solves
u, + H(x. t. u. Du) 2 0 and u(.Y, 0) 2 u(x, 0) = q(x) Alx/ + B 2 max(Co(lxj
so the values given We next set
by (3.30)
provided
+ 1). C(1 f 1.~1)f a((1
that
+ T)A)),
lvork.
z(.r, .“, f) = 11(x, f) - 110’. I) which.
by lemma
1 and (HZ).
is a solution
of
z, + H(.u. ):. D,,?, D,.z) S 0 on {: > 0}
(3.22)
where ti(x, y. p, q) = inf{H(x, satisfies (HO)-(H3). on the form
r, r.p)
- H(,v. r. r. -4):
We seek some solutions
u of U: J_ fi(s.y.
u(x.y,r)=(l
where
+r)(A~s-yi:‘+
A, B 2 0 and 0 < y S 1. If ;I = 1 it suffices
r E R. f E [O. ir]}
D,ru. D,u) 2 0 on {.Y - J; < I}
B)
to have
A1.u - yj -t B T=C(1 + Ix - ~1) + 2a((l
(3.3)
+ T)A).
(3.24)
On existence
and uniqueness
of solutions
of Hamilton-Jacobi
equations
367
since C(1 + /X - yi) + 2a((l
i Z-)A) 3 H(y, t, r, -(l
- H(x, r, r, (1 + t)AD,/x
+ t)AD,lx
- yl)
- yl).
We also have u(x, y. 0) 2 U(X,0) - u(y, 0) if A/x - yi + B 2 C,,(l + Ix - yl). By comparison we conclude that the second inequality of (3.19) holds. In the case y < 1. the inequation is satisfied by u on {lx - y/ < 1) if Alx - y’/’ + B 3 m &l
+ T)yAIx - yIY + 11 - y/)
(3.25)
while D(X,y, 0) 2 z(x, y, 0) holds on {lx - yl < 1) if Alx - ~1;’ + B 2 m,(lx
- yi) on {lx - _v < 1)
(3.26)
and, by (3.19), u(x, y, f) 2 u(x, f) - u(y, t) on {Ix - yl = 1) X [0, T] if A+BsA,+B,.
(3.27)
We achieve all these conditions as follows: put A = max(A,
which guarantees
+ B,, mdl)
+ 1, m,(l)
(3.28)
+ I>,
(3.27), and then
B = K(y) = max{m.((l
+ T)yr + (r/A)‘!‘)
- r) + m,((r/A)l:‘)
- r: 0 =Sr s A}.
(3.29)
By lemma 1 K(O+) = 0
(3.30)
and so the estimate u(x)-u(y)SAlx-ylY+K(y)forlx-yl
(3.31)
provides the desired modulus m. We turn to the estimate of the modulus in t. Here we will depart more dramatically from Ishii’s analysis in order to obtain some new information even in the case he treated. (W’e leave it to the reader to adapt Ishii’s analysis to the current case.) To understand what follows, let us consider two extremely simple cases of the problem II, + H(x, t,
CL, Drc)
=
0 on W” X (0, T] and u(x, 0) = q(x) on R’.
(CP)
In the first case, put H(x, t, p) = f(x). The solution of (CP) is then U(X, t) = q(x) + $((x). If f is unbounded, then u is unbounded and CLis not uniformly continuous in t uniformly in x. In the second case, put H(x, t, rr,p) = u. Now the solution of (CP) is U(X, r) = e-‘q(x). If cp is unbounded then u is unbounded and u is not uniformly continuous in t uniformly in X. However, these cases illustrate the only way in which unboundedness and lack of uniform continuity arise. This is the main content of the proposition we will prove next. This proposition involves the auxiliary function U(x, r) which is defined by letting r--* U(x, t) be the solution of the following ODE initial-value problem: Ii’ + H(x, t, Ii, 0) = 0, qx,
t) = q(x),
(3.32)
368
M. G. CRANDALL and P.-L. LIOM
where ’ means *.d/dr”. The continuity of H and the monotonicity of H(x. r, U, 0) in I/ guarantee that U is well defined on R” x [0, T] and it is easy to see that U E UC,(R~” x [O. T]). Since U is a viscosity solution of U, + G(x, t. U, DU) = 0 where G = H(x, t. U, 0) satisfies the necessary conditions, we deduce from the above that IU(x, [)I s A& with constants
from (3.20).
From
(3.33)
(3.33)
+ Bo
and (3.32) we conclude
that
iU’(x. r)i < F(R) for 1x1< R
(3.34)
so U is Lipschitz continuous in t uniformly for bounded x. Thus the assertions concerning modulus in t locally in x of theorem 4 follow from the part of theorem 4 already proved
the and:
1. Let the assumptions and notation of theorem 4 hold. Let lJ be defined by (3.32) where u(x. 0) = q(x). Then there is a function g: [0, T] - W with g(O+) = 0 and depending only on C from (3.1), (3.2), C,,, rnq (from (3.17)), m, (from (Hl)) and o (from (3.3)). such that PROPOSITION
ilc(x, t) - U(x, t)] c g(f) forx E RV, f E [0, T]. Remark 6. The proposition suffices to establish sided continuity is a uniform two-sided continuity at t = 0 may be repeated at any f > 0. Proof.
By lemma
2 and (H2).
(3.35)
the local modulus in t since a uniform oneand the continuity estimate of the proposition
Z(X, y, r) = U(X, 1) - U(y, t) is a viscosity
solution
z, + fj(x, y, Dxz, D,z) s 0 on {z > 0},
of (3.36)
where H(x, y, t, p, q) = inf{H(x, satisfies
(HO)-(H3).
In the usual
t, r. p) - H(y. t. r, 0): r E R, f E [0, T]}
way, if Ao, B. are given by (3.20)
iu(x, t) - U(y, t)j s (1 + I) (A& because the right-hand side is a supersolution on Ix - y/ < 1 in the form u(.r,y, where
- y/ + B,)
(3.3s)
of (3.36), etc. Now we seek another
r) = (1 + t)A(Ix - _tI: + E)‘~ + tB + D
A, B, D a 0, 0 < y s 1 and E > 0, which further u(x,y.
(3.37)
supersolution
(3.39)
satisfies
0) 2 mo(~.r - yj) ?= u(x, 0) - U(y. 0) = q(x)
- q(y)
(3.-lO)
and u(.Y, y, f) 2 (1 + T)(A,, + Bo) 2 u(,u, t) - U(x. f) if 1.~- y/ = 1. If we achieve
all these
things,
then
n(.r, t) - L;(.u. 1) < u(.T, s, t) < (1 + r)A&:’ 2 L tB + D by comparison.
(3.41)
(3.11)
On existence and uniqueness of solutions of Hamilton-Jacobi
equations
369
We leave it to the reader to verify that (3.41) holds for arbitrary B, D 2 0 if A = max((l
+ T)(A,
+ B,,),m,(l)
+ 1).
(3.43)
and then, with A now fixed by (3.43), (3.40) holds if D = K,(y)
= max{(m,((r/A)ii’))
- r: 0 G r CA}.
(3.44)
Finally, we use that ID,A(Ix - y(’ + e)y”/ G Ay/( .E(‘-‘)/~) to conclude that u is a supersolution on /x - yl < 1 provided B = K,(y, E) = a(Ay/(~(~-r)‘~))
+ m,(l).
(3.45)
But now we are done: A is fixed, D tends to zero with y and so the right-hand side of (3.42) can be made as small as desired by taking E small, then y small and, finally, f small. A lower bound is produced in the same way. Remark
7. Using a similar (but somewhat more complex) argument one can prove u - U E BUC(&!’ X [0, T]). In particular, if H(x, t, K, p) is independent of u, the above proof adapts to show that f u(x, t) - q(x) + H(x, t, 0) dr I0
is uniformly continuous in t with a modulus depending only on the usual data. The analogue for (SP) of these results is that if V(x) is defined implicitly by the equation V(x) + H(x, V(x), 0) = 0, and u is the solution of (SP), then u - U E BUC(b!‘). Existence
for (CP)
Theorem 4 and the same device which was used to establish existence for (SP) succeed here, and we need not repeat the argument. Remark
8. The existence results can be used together with remark 5 to strengthen the comparison results when R = R”. We illustrate this for (SP)-the analogous remarks hold for (CP). Let HE C(R” x W x R’) satisfy (HO)-(H3). Let u, u E C(R”) be a sub- and a supersolution of (SP) on W>”such that there is a constant K for which
lu(x) - u(y)1 + lu(x) - o(y)1 S K(l + Ix - yl) for x, y E W”. Then u G u. Indeed, let w E UC(WN) be the solution of (SP) provided by theorem 2. By remark 5, u G w and w < u, so the assertion is correct. Remark
9. The conditions (H3) and (H3)’ are not “natural” in the sense that they are not invariant under simple changes of variables. Hence, for example, they are not sensible on manifolds. To extend the scope of our results to include manifolds, we will formulate a generalization of (H3), and to this end let us consider the following properties of a metric (x, y) + d(x, y) on R’:
370
Lf. G. CRANDALL and P -L. Lto\s
(dl)
there
are constants
jL, A such that
(d2) for x, v E RV and 2 f _v. the map xd(x, y) is differentiable at z. We will denote the derivative of the map xd(x.v) at .K and the map yc/(x.J) at J’ b) d,(x, y) and ci,(x. y) respectively. The generalization of (H3) is: (H3)” there is a distance d satisfying (dl) and (d2) and a modulus mH such that for all (t, r) E [0, T] x 2, x, y E Rv and t 2 0 H(x, t, r, Ed,(x.y))
- H(y,
I, r. -;‘d,(.r.y))
c -mH($
- !I f
I -,L,).
The uniqueness arguments go through if (H3) is replaced by (H3)” when one replaces !.Y- ,V by d(,r,y) at appropriate points in the proofs. Moreover, one can work more generally on manifolds. The extension of the existence assertions to manifolds deserves more than a remark and will be considered elsewhere. ;IcknoMledgemenrs-The
paper. Ons of the authors and partially
supported
authors
are grateful
to H. Ishii for pointing
(MGC) was sponsored by the United by the National Science Foundation
out a shortcoming
States Army under Grant
in a previous
under Contract 60. So. &lCS-SOO3946.
draft
of this
D.A.AG79-SO-C-OOJl
REFERENCES 1. BARLES G.. Controle impulsionnel deterministe, inequations quasivariattonelles et equations de Hamtlton-Jacob] du premier ordre. these de 3’ cycle. Lniversite de Paris-IX-Dauphine (1983). 2. CRANDALL M. G., EVASS L. C. Sr LIONS P. L., Some properties of viscosity solutions of Hamtlton-Jacobi equations. Trans. Am. marh. Sot. 282. 487-502 (1984). 3. CRANDALL ICI. G. & LIOSS P. L.. Viscosity solutions of Hamilton-Jacobi equattons. Trruns. Am. ntarir. Sot. 277. l-42 (1983). 4. CRANDALL .LJ. G. & Lro,us P. L., Solutions de viscosite non bornees des equations de Hamilton-Jacobi du premtcr ordre. C. r. hebd. st%v~. Acad. Sci. Paris 298, 217-220 (1981). 5. CRANDALL hf. G. & LIONS P. L.. submitted for publication. 6. CRANDALL hf.G. & NEWCOMB R.. Viscosity solutions of Hamiiton-Jacobi equations at the boundary, Proc. Am. marh. SOC. 94, 283-290 (1985). 7. CRANDALL M. G. 8r SO~KANDIS P. E., Developments in the theory of nonlinear first order partial differential Alabama (1983). equations. Proceedings ofrhe Ituernational Symposium on Differential Equations, Birmingham. North-Holland Mathematics Studies 92, .Uorth-Holland. Amsterdam (1984). 8. ISHIIH., Remarks on the existence of viscosity solutions of Hamilton-Jacobi equations. Buil. Fat. Sci. Engng. Chuo liniu. 26, j-21 (1983). Y. JSHII H., Uniqueness of unbounded solutions of Hamilton-Jacobi equations. Indiana Uniu. .Warh. J. 33, 731-718 (1984). 10. LIONS P. L.. Generalized solutions of Hamilton-Jacobi equations. Research !Vofes in ,Lfarhemarics 69. Pitman, Boston (1982). 11. LIONS P. L., Existence results for first-order Hamilton-Jacobi equations. Ric. .Cfat. ,\‘npoll 32. 1-23 (1983). 12. SOUGANIDIS P. E., Existence of viscosity solutions of Hamilton-Jacobi equations, J. dift Eqns (to appear).
AND UNIQUENESS OF SOLUTIONS JACOBI EQUATIONS
OF HAMILTON-
MICHAEL G. CRANDALL klathemarics Research Center. University of Wisconsin-Madison. !vLIadison.Wisconsin 53706. U.S..1
and PIERRE-LOUIS LIONS Universite de Paris-IX-Dauphine.
place de Lattre-de-Tassigny.
75775 Paris Cedex 16. France
(Receioed in revised form 20 June 1984; received for publicarion j June 1985) Key words and phrases: Hamilton-Jacobi equations, first order nonlinear partial differential equations. existence, comparison theorems, viscosity solutions. moduli of continuity.
ISTRODUCTION OCR MAIN purpose here is to establish some uniqueness and existence theorems for HamiltonJacobi equations. We will focus on the Cauchy problem
U, + H(x, f, U, Du) = 0 in Rv X (0, T). u(x, 0) = f&),
(CP)
and the stationary problem u + H(x, ~1,Du) = 0
in R.v.
(SP)
in which H is a real-valued function of its arguments, x denotes points in 5&l’,Du stands for the spatial gradient (llX1, . . . , u,,~) of 11,which is itself a real-valued function of either (x, t) or x as appropriate. The uniqueness follows from comparison theorems of maximum principle type. The comparison theorems we present here correspond to some results proved in special cases by Ishii in [9] and used by him without proof in more general cases in [8]. These results concern the comparison of solutions which are uniformly continuous in x-but perhaps unbounded-in cases where the Hamiltonian H is restricted by appropriate continuity conditions, but not otherwise by growth conditions. We will give complete proofs in a natural generality which strictly includes the corresponding statements of [8]. Moreover, it would seem to us worthwhile to present our complete proofs even in the special cases used in [9]. For other sorts of uniqueness results for unbounded solutions see Ishii [9] and Crandall and Lions [a, 51. As regards existence, we will prove existence results in the same new generality for which we establish the uniqueness. The main tool used for this is estimated on the modulus of continuity of solutions of (CP) and (SP). These estimates are obtained by using the comparison results and modifications of the elegant ideas of Ishii [8]. The notion of “solution” of the differential equations in (CP) and (SP) that we will employ is that of a “viscosity solution”. We will not explain this notion here and refer the reader instead to Crandall, Evans and Lions [2] for easy access to definitions and some proofs of the 353
sort presented here in the simplest situations and to Crandall and Lions [3]. where v-arious equivalent notions of viscosity solutions were presented and the first uniqueness proofs were given. The question of existence of solutions of Hamilton-Jacobi equations has a long history,. We refer the reader to the book Lions [lo] for references to the substantial ~.ork which predates the notion of viscosity solutions. Existence theorems in the general spirit of this paper were established in model cases in Crandall and Lions [;I. Considerable generality, was achieved in the results of Lions [lo. 111 (’m which boundary problems were considered-vve do not discuss existence under boundary conditions here, although it is obvious that one could). some new results (in 2-v) were obtained bv Souganidis [13] and then arguments were introduced in Barles [l] which brought the existe-nce theory to a new level of generality. Ishii [S] (using estimates of moduli) and Crandall and Lions [4] (relying partly on Barles’ methods) were the first works concerning the existence of (possibly) unbounded viscosity solutions. The rev,ieu paper [7] outlines other aspects of the theory of viscosity solutions and contains a moderateI>, current bibliography. The test begins with a Section 1 which is devoted to preliminaries and statements of the main existence and comparison theorems. The comparison results are then proved in Section 2. Section 3 is devoted to studying the moduli of continuity of solutions of (CP) and (SP). The results on the moduli of continuity quickly imply the desired existence theorems. 1. PRELI.LIISARIES.
NOTATION
AND
STATEXLENTS
OF RESCLTS
We will use the following notational conventions throughout. The set of continuous functions which map a metric space 0 into the reals will be denoted by C(Q) and the subspace of C(Q) consisting of uniformly continuous functions will be denoted by UC(Q). Functions of modulus of continuity type will be denoted by the letter m as n.ell as m nith various subscripts (e.g. m R. mH. etc.). Such functions m map [0, -/-) into [O. x) and satisfy: (i) (ii) (iii) (iv)
m is continuous and nondecreasing; m(O+) = 0: m(n + b) S m(a) + m(b) for 0. b 2 0; m(r) S m(l) (r f 1) for r 3 0;
(1.1)
where (iv) is in fact a consequence of (i) and (iii). We will call any such function “a modulus”. When Q is a subset of F?.‘, UC,(Q x [0, T]) denotes the space of those 11E C(Q X [O. T]) for which there is a modulus m and an r > 0 such that lU(X, t) - u(y. t)l G m(]x - y]) forx, Finally,
in what
follows,
y E R, ].r - y/ s r and f E [0, T].
T > 0 and BR(z) denotes
the closed R-ball
centered
at 2 E 62”.
will typically
be restricted
BR(Z) = {X E 5Qv: /x - z] =z R}, while intB,(z) the following
is its interior. The Hamiltonians set of conditions:
(HO) HE C(lQ’ x [0, T] x II2 x @j; (Hl) for each R > 0 there is a modulus x
we will consider
mR such that for all (x, t,~),
BR@)
jH(x. f. r, p) - H(y, f, r, q)l 5 m,(b
- q! + 1~ - :I);
by
(y, f, 4) E 2’ X [O, rl
On tx~stence
and uniqueness
oi solutions
of HamIlton-Jacob1
equations
35
(HZ) H(x. r. r. p) is nondecreasing in r for all (x. r. p) in 8.v X [0, T] x A’: (H3) there is a modulus mH such that for all (t, r) E [O. T] x 72 and all X. )j E 2v and E.2 0 H(s. f. r. E.(x - y)) - H(y. I. r. 1.(x - y)) 3 -m,(+ Remark 1. The assumption
(H3) is a replacement
- 1’1’ + ix - !I).
for the stronger requirement:
(H3)’ there is a modulus mH such that for all (t, r.p) E [O. T] x A x 2’ and all x. .” E 2’ lH(x, f, r-p) - H(y. f, r,p)i c mH(lx - .4(1 + lpi)): which was introduced in [3] and has been used since then. The significant observation that the original proofs of uniqueness go through under (H3) (although one should choose q to be radial in, for example, the proof of [3. theorem 11.11, as is done later in [3]) is due to R. Jensen who was led to this remark by considering the uniqueness question for Hamilton-Jacobi equations in infinite dimensions in an ongoing investigation. It is easy to see that (HO)-(H3) is strictly weaker than (HO), (Hl), (HZ), (H3)‘. For example. if N = 1, H(x. r.~) = b(x)p satisfies the weaker set of conditions if and only if b is bounded and uniformly continuous and b(x) + cx is nondecreasing for some c, while the stronger conditions replace this last requirement with Lipschitz continuity of b. The analogous remark holds for general N. (Later vve will comment on how (H3) itself may be significantly relaxed.) Finally, the requirement that mH be a modulus may seem restrictive, but it is not-if this estimate holds with any nondecreasing function tending to 0 at O+, then it holds with a modulus. The uniqueness and existence results for (CP) are: THEOREM 1.
Let H satisfy (HO)-(H3).
Then:
Comparison. Let R be an open subset of iw”, 0 its closure and aS2 its boundary. o E UC,r(5? X [0, 7’1) satisfy the differential inequalities
Let U.
Il, + H(x, f, u, DU) G 0 and 0 < u, + H(x, f, u, Du) on R X (0, T) in the viscosity sense. Assume also that u(x, 0) 6 u(x, 0) on R and u(x, t) s u(x, t) on aR x [0, T].
(1.2)
Then 14c u on R x [0, T]. Existence. Let ~j E UC(IB”). Then (CP) has a solution u E UC,r(W*vx [0, r]) in the viscosity
sense. Remark 2. Theorem 1 extends, via the substitution u = e -“u, to Hamiltonians H(x, r, u,p) + Arc satisfies the assumptions in place of H for some A.
H for which
Of course, the “comparison” theorem when formulated in this way gives uniqueness and much more-it also implies continuous dependence and will be used to estimate moduli of continuity, etc. The parallel theorem for (SP) results upon interpreting (HO)-(H3) for a
356
>I.G. CR.GDAL.Lend P.-L.LiO\j
function H: X,v x R x R.v+ 2, bv regarding considered above. We have: 2. Let H E C(2’
THEOREM
Comparison. inequalities
Let
Has a (r-independent)
special case of the functions
x 2 x Zv) satisfy. (HO)-( H3). Then:
Q be an open
subset
of 2’
and
M. u E UC’(o)
satisfy
the differential
u f H(.r. II. Du) < 0 and 0 G u + H(.r. ~1.Du) on 0 is the viscosity Exisrence.
sense and II 4 u on 3Q. Then
(SP) has a solution
(1.3)
II s u on 0,
II E L’C(;;iL).
Remark 3. Assume that dR is of class C2 and v, denotes the inuard normal to R at .YE 3Q. Then the assumed comparison u G u of aQ x [O. T] in theorem 1 and on 6Q in theorem 2 can be weakened by replacing dR by the set of .r E aQ for which A - H(.u. f. II. p L Av,) is not nonincreasing for some (f. ~1.p) E [O. T] x 2 x 5%”and \,ariants of theorems 1 and 7 remain true. One needs to strengthen (H3) near the boundary a bit-see Crandall and Nexvcomb [6]. The next lemma collects some simple estimates used throughout the test for con\,enient reference. It may be skipped at this time. LE~IXIA 1. Let
~FZ
be a modulus.
Then:
(i) Let E, 6, r > 0. m(1) > 0 and c’,+ nz(r) 2 r’/‘s. Then. r G 7(r~z(l)~)’ (ii) If E 2 0, F>
’ andr’;E
nz(1) and k’(;f) is defined
s a f
Proof.
rF1(?(VZ(1)&)’
‘).
small. (1.4)
for ;/ > 0 by
K(;j) = sup{vr( yEr -I- (r/F)’ then K(O+)
if E and b are sufficiently
1’) - r: 0 s r s F}
= l,ifn K(;/) = 0.
We first sketch
the proof
r2 S e(m(l)(l
of (i). The assumptions
and (l.l)(iv)
+ I) f S) =5 4E mas( d. m(l).
yield
m( 1)r)
and. choosing the various possibilities for the mas on the right. the Lvorst estimate on r uhich arises for small 8, E is the one claimed in (1.1). Then the second estimate arises from the monotonicity of m. (Obviously these estimates can be improved. but \ve \\ill not need to do so.) We just sketch the simple proof of the (aivkward looking) assertion (ii). Let I’.. be the maximum point of f:,(r)
=
TFZ(
;.fEr f (r/F)’
:‘) - r
2 m(O) - 0 = 0. If (r:./F)’ z’has a limit point 8 as ;’ ; 0 and 0 < 8. then on [0, F]. Clearlyfy(r,) the corresponding limit superior of f:,(r,) is at most nr( 1) - F < 0 (so this cannot be). ivhile if B = 0 then the limit superior is at most m(O) - 0 = 0. Since K(;./) =f.,(r..). u.e are done.
We conclude this section with a useful lemma which illustrates the advantage one can take of the linear dependence on the time derivative in (CP). This is a convenient packaging of a standard argument. (For a sketch of a proof in a special case see Ishii [8, lemma 3.31.) LEMXA 2. Let R be an 0 E C(Q x (0. 7)) satisfy
open
of w‘“, H, G E C(Q x (0, T) x R x RL)
subset
and
U.
LI, + H(x, t. II, Du) s 0 and 0 G u, + G(x. f. 0, Dv) on R x (0, r). Define
2(x. y. r) on R
x R
(0. T) by z(x, y. r) = U(X, f) - u(y. 0. Then
x
Zl + H(x, f. U(X, f), D,z) in the viscosity
- G(y, t. o(y, t), -D,z)
c 0 on Q X Q X (0, T)
sense.
Proof. It will suffice to show that if q E C’(Q X R x (0, T)) and (x7 y, f) has a strict local maximum qi(“.
4x,
at (a, b, c) E R
t) - u(y, t) - e(x, .l’, t) x R
x
(1.5)
(0, T), then
b, c) + H(a, c, ~(a, c), D,rq(a, b, c)) - G(b, c. u(b. c). -D,q(n,
b, c)) s 0.
(1.6)
Put d(,u, y, t. s) = max(]x - aI, ly - 61, /r- c . IS - cl). Since the maximum such that
is strict,
there is an r. > 0 and HE
U(X. t) - u(y. f) - q(x.y, for 0 < d(x, y. f, f) G ro. We claim: given r > 0 there
qJ(,y, y, t, s) =
has a local maximum
point
E >
- H(d(x,y,t.
t))
(1.8)
0 such that
u(x, t) - u(y, s) -
rp(xzL’? 0 - (t -
d2b
(1.9)
(x,, yr, t,, s,) which satisfies +,,Y,,
Assuming
C([O. r,]) with H(r) > 0 for 0 < r < r.
t) s rc(a,c) - o(b, c) - ~(a. b,c)
is an
(1.7)
the claim is correct
for the moment
t,, J,) 6 r.
(1.10)
we then have.
by the assumptions.
If these inequalities are added, r is sent to 0 and (1.10) is taken into account, It remains to produce (xr, yI, t,, 3,). Fix r C r. and choose E so that ~U(y. f) - U(y, S)i + lU(x, f) - U(X,S)] + ]cp(x,y, t) - V(x,y,
s)i == (t -
s)‘/E
the result follows.
+
H(r)/2.
(1.11)
58
41. G. CR.ANDALL and P.-L. LIOXS
for n(x. y. r. s) c rO. This is clearly possible. Let Y be given by (1.9) and d(x. y. f. s) = r. Then either d(x. _v. t, t) = r or d(.u. ,v, s, s) = r. Assume d(x. y. I, t) = r (the other case is treated a similar way). Then using (1.8) and (1.11) we have V(X,_y. 1, s) 5 U(X, f) - u(,V. t) - q(X,)‘.
f) - (u(,V. 5) - u(_V. f) t (f - 1): !‘E)
s Y(a. 6. c. c) - H(r) - ;oi,V. s) - U(_V,f)/ - (f - $)2;‘& =z Y(a. b. c, c) - H(r)/2. so the maximum of Y over the set (f c r must occur at an interior maximum satisfying (1.10). 7. PROOFS
OF THE CO.LIPARISOX
We will first prove
essential
theorem 1. which is more difficult step in the proof of the comparison assertion
LEMMA 3. Let the assumptions of the comparison bounded from above on R x [0, T]. Proof
of lemma
3. Since
point,
which is then a local
RESULTS
than theorem 2. The next lemma of theorem 1.
assertion
U, u E rlC,,(Q x [0, ir]), there
of theorem
is a modulus
1 hold. Then
is an
u - ~1is
m and an r > 0 such
that lu(.r, t) - ~(y, t)l + /u(x, r) - u(,v, f): G m(lx - J/) (2.1)
for Jc, y E 0 and 1.t.- y] s r. Let n(,u) = distance(x, Since 11< u on aR x [0, T] and (2.1) holds.
(2.2)
JQ).
we have
U(X, t) - u(y, f) G m(lx - yl) + m(min(d(x),
d(y)) f0r.u. ,VE a and
t E [0, TJ such that ]x - yj 4 r and min(d(x),
d(y))
(2.3)
Thus to bound u - u it will suffice to bound end, choose such an a and set
6 r.
~(11. t) - ~(a, t) for n E Q and d(a) 2 2r. TO this
K = 1 + 2m(r)/r’,
Y(x, y, r) = u(x, t) - u(y, l) - K(jx - al? + jy - al’), 44
= supW(x,
y, 4: (x, y> E &((a,
(2.4)
a>)),
for 0 G t < T. Here &((a,
u)) = {(X, y) E W X W: jx - u/l + ]y - a/* < r’} c G x !3
by d(a) 3 r. In order to prove the lemma, it suffices to show that w is bounded from above uniformly in a. Clearly w is continuous. The main part of the proof is to show that w’ is bounded above on the open set {w > O} = {t E [O, 7-l : w(t) > 0)
On existence
and uniqueness
o!’ solutions
of Hamllron-Jacob
equation>
by a constant in the viscosity sense (and hence in the distribution sense-see I. 111). Since u c u on R x (0). vve have
359
[2. proposition
w(0) 6 sup{u(x. 0) - u(1’. 0) - K(Jx - 1111+ I?’ - ai’) : (x. J) E B,( (a. a))} s ‘m(r). If also w’ G C on {w > 0). we conclude that w(l) s 2m(r) + CT on [O. T]. whence the result. It remains to show that w’ is bounded above on {w > O}. The constant K in (2.1) is chosen so as to guarantee that Y(x,_~.r) < Y(a, a, t) if (x._v) E JB,((a. a)). so there is a point (,f. ,V)E int B,((a, a)) such that w(r) = u(l, f) - u(,:. f) - K(1.f - 1211+ ~!:- alZ).
(2.5)
Of course, f-9 depend on t and u. Let I be an open interval in {IV.> 0). p E C’(I) and w(f) - p(f) have a local maximum at TV I. Let (2.9) E intB,((cr. 0)) be such that (2.5) holds with f = i. Then for all (x, y. f) near (_c.j. r) U(X, r) - u(y, f) - K(jx - 0/Z - ly - a]‘) - p(r) 6 u(X. ?) - u(_V,S) - K(1.f - ui2 - I? - 01’) - p(S). It then follows from theorem
1. lemma 2 and the definition of viscosity solutions that
p’(i) + H(2. I, ~(2, i), 2K(,r - a)) - H(y. i, u(y. i). - X(y
- u)) s 0.
(2.6)
We will use this with the estimates given next to reach our conclusion. Clearly ZKI,? - a]. 2K]j - a/ G 2Kr.
(2.7)
The string of inequalities written next will be explained immediately H(P, i, u(f, F), 2qx
- cl)) - H(j) I, u(j, i). -2K():
below.
- a))
2 H@, F, U(f, T). 2K(f - li)) - H(_V.i. u(.\-, i). -2K()i - a))
(2.8)
- y/ + 2Kj,f - a] - 2Kly - a])
2 -rn2& 2 -mzK,(2r(l
+ 2K)).
The first inequality is due to the assumption w(f) > 0 (which implies n(X. i) > u($, r) > 0) and (H2). In the next inequality, we use (2.7) and (Hl). The final inequality uses I.\:- ?_Is 2r and (2.7). From (2.6) and (2.8) we now know that, in the viscosity sense, w’
ZKr(2r(l + 2K) on {w > O}.
and the proof of lemma 3 is complete. Remark
4. The reader may observe that only (HO)-(H2) were used in the proof of lemma 3 and (H3) was not required. An analogous remark holds concerning the study of (SP) later.
End ofproof of comparison fur (Cl’). Assume the hypotheses of the theorem, holds and that 116 u fails. We may choose a constant c such that
c > 0 and M = sup{u(x, t) - U(X,r) - C~:XE n and 0 G t s i? > 0.
that (2.1) (2.9)
360
41.G. CR;\\D-\LL
Note that M < x by lemma
P.-L Llo
and
3. Next let E. @ > 0 and set
y’)).
Y(.r,,V. r) = u(x. f) - u(_V, f) - Cf - (!x - >$ ;t. L i;(‘.V :
(3.10)
Clearly S(P) = M - sup{Y(.r,
(2.11)
x, t) :x E G. 0 G f 9 7-}
satisfies = 0
(2.12)
y. I) :(x,y. t)E fi x fl x [O.T] and 1s - >! < r} 2 :CI - d(p).
(7.13)
d(o+) and sup{Y(.r,
In view of (2. l), for (x. J, r) E d x Ci=2 x [O. T] satisfying
I.~ - J, G I’ vv.e have
M + m(i.U - yi) - (1-r - y/‘/E + /3(i.Vi’ + ~J,,‘)) 12.14)
2 11(x, t) - u(x, I) - cr i (U(.Y. f) - 1:(_V.t)) - (.\’ - .“,yE + /3(1.yi’ + Iyi’)) = Y(x. )‘, f)
and from (2.13) and (2.14) it is clear that Y attains a positive masimum with respect to the set {(I, y, t) E h X G X [O. T] : Ix - ~1 =2r} at some point (.f. s. 7) if /3 is small (as we hereafter assume). We now proceed to make some simple and standard estimates inv.olving (.c. !:. F). Usins (2.1-l) and (2.13) we have (2.15)
6(/j) + m(]r\’ - I_$ 2 11 - _$/E i /4( .f 1 f i\:l’). From
(2.15)
provided
and lemma
1 we deduce
that (taking
m(l)
+ 0 \vithout
(i)
I.?- ,PI G Z(m(l)E)’
(ii)
~- - ,i2!E =s b(P) + m(2(nr(l)E)1 1.r
(iii)
/?(l,tl’ i !?I’) 6 6(/3) i m(?(??Z(
only that E and ,!3 are small.
loss of generalit>)
‘. (2.16)
?). 1)E)' ?).
Next we check that f> 0. Indeed.
Y(x,,V, 0) = u(.u, 0) - u(x. 0) + u(s, 0) - u(_V. 0) -
((is - .‘.l%
the inequality
t &r
1 - !I:,)
d m(1.V - J’) - (‘.V - viz/E + P(I.yl’ + lvi’)) coupled
with (2.16)
shovvs that Y(_f.,7.0)
so Y(2.L;. 0) 3 M - S(/3) is impossible or d(y) s r. (2.3) yields
G rn(Z(VZ(l)E)’
if E and p are small.
‘) Thus r> 0. Similarly.
if ti(.i-) d r
Y(1. F. r> < 2m(r). nhich cannot be if r is sufficiently small (and we may take it to be so). Lastl!.. (2.16) (i) guarantees that 1-U- ~~1< r if E is sufficiently small. Thus vve can guarantee that (.i-. .\-. F) is an
On existence
interior
point
and umqusnrs,
of solutions
of Hamilton-Jacob1
361
equations
of the set {(x, Y. f) E G x 0 x [O, 7-l : lx - j/ c r}
for vvhich it provided To proceed. put
a maximum
of Y.
p = 2(X - )s)/E, qx = 2p,e, qv = -2py. Since Y has an interior solution yield
maximum
point
at (,?,L;, Q, lemma
(2.17)
2 and the definition
1. u(i. r,>, p + qx) - H()s, I. o(Y, i),p
c + H(i,
+ qJ
We next write a string of inequalities to be used in conjunction them immediately following: We have t‘), p + q.t) - H(j,
H&f. r, 42,
I, u(j, g,p
I, IQ. S),p + q*) - H()?, I, u(Y, i),p
a H(Z
i, u(j,
0, p) - H()?, i, u(g, 0.P) -Pi
The first inequality is due to the monotonicity (H2) inequality arises from (Hl) when R, denotes a bound from (2.16) and (2.17) that
for a suitable
constant
(2.18)
with (2.16) and then explain
+ qJ
(2.19)
- (m,,(lq,l)
!P + /If - r‘j) + mR,(lq,zI)
(d&)jpl
s 0.
+ qr)
2 !f(x,
2 -(m,(!*f
of viscosity
+ m&(lqj
)I>
+ m&?,J)). of H and u(X,?) > o(y,7). The second on ipi, (p + qll, and lp + qJ. It is clear
(2.20)
c C and ]qX1,/qv/ s Cd/p
C as E and /3 tend to 0. Thus for some C we have R, s C/~/E.
(2.21)
The final inequality of (2.19) arises from (H3) and p = 2(Z - Y)/E from (2.17). again we have 11 - y] + 0 and /,r - y/ ip] = 211 -$/E
< 26(P)
+ m(2(m(l)&)r
Using
2).
(2.16)
(2.22)
We are now essentially done: letting /3 J 0 and then E J 0 in (2.19) yields 0 on the right-hand extreme of (2.19) because of (2.20) and (2.22), so (2.18) implies that c G 0, a contradiction. Proof of comparison for (SP). The proof of the comparison common with that for (CP). and where it differs it is simpler. theorem 2 we first prove that: LEMXA 5. Let 11and u be as in the comparison from above. Proof. Let m be a modulus :G)
- 4Y)l +
assertion
result for (SP) has much in Assuming the hypotheses of
of theorem
2. Then u - u is bounded
and r > 0 be such that
14~)-
4Y)l < m(]x - yl) for x, Y E 0 such that IX - y] s r.
(2.23)
362
>I. G. CRUDALL
Analogously
and
P.-L. Lro~s
to (2.3) we have [l(x) - o(y) s m(lx - yl) + m(min(d(x).
d(y)) f0rx.y
such that ~,r - y! G r and min (d(x),
s r.
E B (2.24)
Choose
a point
d(y))
a E 0 such that (7.35)
u(a) - u(a) > m(r), (if there
is no such a we are done) K = 1 + Zm(r),‘r’
and
and then d(a) > r by (2.21). Y(x,y)
The function
Y defined
= U(X) - u(y) - K(1.r - u12 i iv - u;‘).
by (7.26)
satisfies Y(u, a) > Y(x, y) for (x, y) E aB,(u, a) and so it will have an interior maximum (~I?,?_) with respect to this set. We then use the definition of viscosity sub- and supersolutions to find that u(X) - o(j)
+ H(1. u(x), 2Kc.r - u)) - N(y, u(_G). -2K(,?
- u)) s 0.
Since u(i) 2 u(y) (by (2.23) and Y(,f, j) 2 Y(u, u)) we use (Hl)-(H2) theorem 1 to conclude that ~(3) - u(y) is bounded above and then U(U) - u(u) 4 u(i)
now choose
of
- u(V) + 2m(r)
to conclude that u - u is bounded above. To complete the proof of theorem 2, we assume &I = sup(u(x)
as in the proof
that (7.27)
- u(x)) > 0:
E > 0 and /3 > 0 and let (x, I;) be a maximum Y(X, ,v) = U(X) - u(y) - (is - $/&
point
of
+ $0(1X1”+ IylZ))
over the set S = {(x, y) E 0 x G: Ix- y/ =Zr}.which exists since M + m(lx - yi) - (IX -y/‘/E
+ p(iXl’ + iyi’)) 2 Y(X,,V)
(2.28)
on S. Moreover, (2.29)
Y(X, Jq 2 IV2- S(P) where S(O+) = 0. As before, assumptions, if
one
rules
out X, jt E a.S for 6, E, r small
and then,
using
the
p = 2(P - y), qx = 2@, qy = -2py we have &I - c%0 =Z U(X) - u(T) 6 H()?, u(,‘>,p
+ qy) - H(&
u(.f),p
+ qr).
Finally, arguing as in the proof for (CP), lpi is b ounded by C/d& for small E and p, while with E fixed q,c and qv tend to zero like g/p and the proof is completed just as before. Remark 5. One knows, in dealing with bounded solutions, that uniform continuity of u or u (and not necessarily both) suffices to prove comparison results. The analogous statement in
363
On existence and uniqueness of solutions of Hamilton-Jacob1 equations the present of theorem such that
context is that the above proofs easily adapt to prove that the comparison assertions 1 remain correct provided only that u and u are continuous, there is a constant K lU(X, r) - rc(y, f)/ + 10(x, r) - o(y, t)! s K(1 + 1.r- yl)
for (x. t), (y, t) E 0 X [O. T] and one of 11, u lies in UC,,(a x [O. 7-1). The analogous statement for (SP) arises upon lettin g u and u be independent of t. Later we us2 this remark and the existence results of Section 2 to improve the comparison results in the case R = 2.v. 3. LIIODULI OF CONTINUITY XSD EXISTENCE Throughout this section we assume that H satisfies (HO)-(H3) in the case of (CP) and. as before, interpret these conditions and others to be laid below in the case of (SP) in the obvious ways. To begin, following Ishii [8], we label the following structure properties of H: C will be a constant such that ]H(x, t, 0, O)i G C(1 i 1x1)
(3.1)
jH(x, f, U, 0) - H(y, t, U, O)] 6 C(1 + 1.X- _vl)
(3.2)
or for X, y E iRv, K E R, t E [O, T]. The existence of such constants C is guaranteed by the assumptions. Contrary to the previous section, we will begin with the simpler case of (SP) and prove: THEOREM 3. Let HE modulus m depending
C(R” x W x R”) satisfy (HO)-(H3). only on C,
If also (3.2) holds,
then
there
a(R) = m,(R). (where mR is from (Hl)), of (SP) then
and mH (from (H3))
(3.3)
such that if u E UC(R”)
is a viscosity
solution
]u(x) - u(y)1 G m(lx - y]) for Js. y E R’. Moreover,
if (3.1) holds,
is a
(3.4)
then ]I+)]
G C(1 + Ix]) + a(C).
(3.5)
Proof of theorem 3. We begin with the estimate (3.5). The proof, given our comparison result, is essentially the same as the proof of [8, proposition 3.21. The only property of c~ needed is ]H(x, t, u,p) In order
to guarantee
- H(x, t, U, O)i < a(R) for lpi s R.
that u(x) = Alxj + B,
where A, B s 0, solves
u + H(x, U, Du) 2 0, we observe
that
u + H(x, u, Du) 2 Alxl + B + H(x, 0, ADlxl) 2 Ajxl + B + H(x, 0.0) - a(A) 2 Alxl + B - C(l + Ix]) - a(A),
(3.6)
and then take A = C and B = C + a(C). (We are being a little bit formal: how,ever. the subdifferential of ix; at the origin is the unit ball, and so D1.r; stands for an arbitrary v’ector in the ball at x = 0. and the computation is valid.) The estimate (3.5) with II in place of ~II~now follows from comparison. One then estimates u below by -U in a similar vvav to complete the proof. Kext, vve can prove in a similar way that if C is from (3.2). then U(X) - u(y)’ s qu Indeed,
observe
(3.7)
- I‘~ + (C + Za(C)).
that .?(X.!‘) = U(X) - u(y)
is a viscosity
solution
of 7 + H(X, U(X). D,i)
and so the monotonicity
property
z(.~,y)
i
Drz)
U(J). -D,.Z)
(fG,
=
0.
that
- H(y. I.
Ju _ ,xz.v x @v, H(,~,L’,P. 4) = j$
(3.9) and (3.10)
- H(y.
of H guarantees
f H(x, u(y),
DefineH:~~x~.Lx~).~xD~i A
Clearly
(3.8)
-D!z)
s 0 on (2 > 0).
(3.9)
R by 7,~) - NJ.
(3.10)
r, -q)).
impI) -7 + ti(x, y, D.,z, D,.z) G 0 on {z > O}.
and it is also obvious
that fi satisfies
(HO)-(H3)
on %’
(3.11)
X z’.‘. We seek solutions
u(x, y) = A!x - ~‘1’ + B
(3.12)
of u + fi(x,y, D,ru, DLu) 2 0 on Xv,
(3.13)
where A, B 2 0 on 0 < y 4 1. We make some formal (when x = y) estimates below which one can (following Ishii) make rigorous by first replacing js - y!? by (ix - yi2 T E)” ‘) and then letting E-+= 0. We estimate for all r: u + H(.r, r. D,u)
- H(y. r, -D,u)
+ H(x, r, :/AI-u - yiY-‘(x
s Ajs - ).I? + B
- y)] - H(y,
In the case y = 1 we use (3.2) and (3.6) to estimate
(3.14)
r, ;/Alx - yl’-‘(I the right-hand
- y)),
side of (3.14)
below
b>
Alx - yi + B - (C( 1 + /X - yi) + 20(A)) so u satisfies
(3.13)
if A=CandB=Za(C)+C.
Since z < u on a{z > 0}, we use the comparison result for (SP) to conclude that (3.7) holds. We will use the bound (3.7) while estimating the modulus of continuity. Let ;J < 1. We will
On existence and uniqueness of soluuons of Hamilton-Jacobi
equations
compare z and u only on the set {Z >
0)n {IX - ~1< 1)
sowe want z 6 u on the boundarv of this set. Bv (3.7) this will hold if (3.15)
A 3 2c + 2a(C). We guarantee
this by setting, for reasons which will soon be evident. A = max(2C + 2a(C). mH(l) + 1).
(3.16)
Next we use (H3) to estimate the right-hand side of (3.13) below by Alx - yl’ + B - m,z,(yAlx - _v/:’+ jx - _v/)
and, thinking of Ajx - yl’ as r, we see this is nonnegative B 2 K(y) = max{mH( yr + (r/A)“‘)
on {]x - y] < l} provided that - r: 0 G r s A},
Since A > mH(l) by (3.16), lemma 1 yields K(O+) = 0. We now have shown that U(X)-u(y)sK(y)+Alx-y]YforO<;~~land]x-y~
of y and holds. Thus m(r) = inf{K(y) + Ar’: 0 < y =S 1)
provides the desired modulus on [0,11. Proof of existence for (SP). It follows from theorem 3 and the uniqueness that if {H,,} is a sequence of Hamiltonians on w” X R X w” satisfying (HO)-(H3), (3.1) and (3.2) uniformly (with th e same ma, mH, C), &, E uC(Rv) are solutions of 11, +
H&c, u,, Dun) = 0
and H, tends locally uniformly (on R” X W X IRS) to a limit H, then there is a solution u E UC(Rv) of u + H(x, u, Du) = 0 and u,,+ u locally uniformly. Thus existence follows if, given H satisfying the assumptions, we can produce sequences {H,} and {u,}. Let 8 E C?(R) be nonnegative, symmetric, 8(r) = 1 for Irl s 1 and 6(r) = 0 for Irj 2 2. Put H,(s. 1t.p) = H(x, u, 0(lpi/n)p). Clearly H, satisfies the required conditions uniformly and also. for fixed n. H, satisfies (HO)-(H2), (H3)‘. Ishii’s existence theory may be invoked to provide the u,, and we are done. We turn to the Cauchy problem. Modulus
of continuity for (CP) Let HE C(R” x [0, T] X R X ET”) satisfy (HO)-(H3), (3.1), (3.2) and (3.6). Let u E UC,(R” X [0, T]) be a viscosity solution of u, + H(x, t, u, Du) = 0 on W” x (0, T]. Let Co
THEOREM 4.
41. G. CRA>D.ALL and P.-L. Lrow
366
be a constant, lq(x)l
q(x)
= u(x. 0) and m, be a modulus
s CO(l + Ixi) and iv(x) S m,(ix
- y1) fors.
such that
- q(y)1 s Co(l i ;.Y- !:I), and iv(x)
- F(J)
(3.1’)
,v E RV.
Then there is a modulus m depending mH (from (H3)) and T such that
only on C (from
(3.1).
(3.2)).
CU. mF (from
iU(X. t) - “(I’, t)l s m(ix - ,11) for x, !: E 52.’ and f E [O. 7’1.
(3.17)).
(3. IS)
iMoreover, Iu(x, f)I G (1 -I- L)(A&
+ B,,) and lu(s, f) - ~(y, t)i 5 (1 + r)(A,l.r
- Jo f &,).
(3.19)
where A,, = max(C.
C,), B0 = mas(Cli.
C + ?a((1
+ ir)A,,)).
(3.20)
Also for each R > 0 there is a modulus mR depending only on C. Co, ~2,~. u. mH. T and the function F(R) = max{lH(z. r. r. O)l: /=I s R. f E [0, T], /r/ =GA0R + B,,}. such that ~u(x. r) - U(X, s)l d mR(ir - s/) for .K E BR(0) and f, s E [O. ir]. Proof. The strategy of proof is the same as in the previous of (3.19). one shows that
(3.21)
case. To verify the first inequalit).
u = (1 + t) (Ai,ri + B) solves
u, + H(x. t. u. Du) 2 0 and u(.Y, 0) 2 u(x, 0) = q(x) Alx/ + B 2 max(Co(lxj
so the values given We next set
by (3.30)
provided
+ 1). C(1 f 1.~1)f a((1
that
+ T)A)),
lvork.
z(.r, .“, f) = 11(x, f) - 110’. I) which.
by lemma
1 and (HZ).
is a solution
of
z, + H(.u. ):. D,,?, D,.z) S 0 on {: > 0}
(3.22)
where ti(x, y. p, q) = inf{H(x, satisfies (HO)-(H3). on the form
r, r.p)
- H(,v. r. r. -4):
We seek some solutions
u of U: J_ fi(s.y.
u(x.y,r)=(l
where
+r)(A~s-yi:‘+
A, B 2 0 and 0 < y S 1. If ;I = 1 it suffices
r E R. f E [O. ir]}
D,ru. D,u) 2 0 on {.Y - J; < I}
B)
to have
A1.u - yj -t B T=C(1 + Ix - ~1) + 2a((l
(3.3)
+ T)A).
(3.24)
On existence
and uniqueness
of solutions
of Hamilton-Jacobi
equations
367
since C(1 + /X - yi) + 2a((l
i Z-)A) 3 H(y, t, r, -(l
- H(x, r, r, (1 + t)AD,/x
+ t)AD,lx
- yl)
- yl).
We also have u(x, y. 0) 2 U(X,0) - u(y, 0) if A/x - yi + B 2 C,,(l + Ix - yl). By comparison we conclude that the second inequality of (3.19) holds. In the case y < 1. the inequation is satisfied by u on {lx - y/ < 1) if Alx - y’/’ + B 3 m &l
+ T)yAIx - yIY + 11 - y/)
(3.25)
while D(X,y, 0) 2 z(x, y, 0) holds on {lx - yl < 1) if Alx - ~1;’ + B 2 m,(lx
- yi) on {lx - _v < 1)
(3.26)
and, by (3.19), u(x, y, f) 2 u(x, f) - u(y, t) on {Ix - yl = 1) X [0, T] if A+BsA,+B,.
(3.27)
We achieve all these conditions as follows: put A = max(A,
which guarantees
+ B,, mdl)
+ 1, m,(l)
(3.28)
+ I>,
(3.27), and then
B = K(y) = max{m.((l
+ T)yr + (r/A)‘!‘)
- r) + m,((r/A)l:‘)
- r: 0 =Sr s A}.
(3.29)
By lemma 1 K(O+) = 0
(3.30)
and so the estimate u(x)-u(y)SAlx-ylY+K(y)forlx-yl
(3.31)
provides the desired modulus m. We turn to the estimate of the modulus in t. Here we will depart more dramatically from Ishii’s analysis in order to obtain some new information even in the case he treated. (W’e leave it to the reader to adapt Ishii’s analysis to the current case.) To understand what follows, let us consider two extremely simple cases of the problem II, + H(x, t,
CL, Drc)
=
0 on W” X (0, T] and u(x, 0) = q(x) on R’.
(CP)
In the first case, put H(x, t, p) = f(x). The solution of (CP) is then U(X, t) = q(x) + $((x). If f is unbounded, then u is unbounded and CLis not uniformly continuous in t uniformly in x. In the second case, put H(x, t, rr,p) = u. Now the solution of (CP) is U(X, r) = e-‘q(x). If cp is unbounded then u is unbounded and u is not uniformly continuous in t uniformly in X. However, these cases illustrate the only way in which unboundedness and lack of uniform continuity arise. This is the main content of the proposition we will prove next. This proposition involves the auxiliary function U(x, r) which is defined by letting r--* U(x, t) be the solution of the following ODE initial-value problem: Ii’ + H(x, t, Ii, 0) = 0, qx,
t) = q(x),
(3.32)
368
M. G. CRANDALL and P.-L. LIOM
where ’ means *.d/dr”. The continuity of H and the monotonicity of H(x. r, U, 0) in I/ guarantee that U is well defined on R” x [0, T] and it is easy to see that U E UC,(R~” x [O. T]). Since U is a viscosity solution of U, + G(x, t. U, DU) = 0 where G = H(x, t. U, 0) satisfies the necessary conditions, we deduce from the above that IU(x, [)I s A& with constants
from (3.20).
From
(3.33)
(3.33)
+ Bo
and (3.32) we conclude
that
iU’(x. r)i < F(R) for 1x1< R
(3.34)
so U is Lipschitz continuous in t uniformly for bounded x. Thus the assertions concerning modulus in t locally in x of theorem 4 follow from the part of theorem 4 already proved
the and:
1. Let the assumptions and notation of theorem 4 hold. Let lJ be defined by (3.32) where u(x. 0) = q(x). Then there is a function g: [0, T] - W with g(O+) = 0 and depending only on C from (3.1), (3.2), C,,, rnq (from (3.17)), m, (from (Hl)) and o (from (3.3)). such that PROPOSITION
ilc(x, t) - U(x, t)] c g(f) forx E RV, f E [0, T]. Remark 6. The proposition suffices to establish sided continuity is a uniform two-sided continuity at t = 0 may be repeated at any f > 0. Proof.
By lemma
2 and (H2).
(3.35)
the local modulus in t since a uniform oneand the continuity estimate of the proposition
Z(X, y, r) = U(X, 1) - U(y, t) is a viscosity
solution
z, + fj(x, y, Dxz, D,z) s 0 on {z > 0},
of (3.36)
where H(x, y, t, p, q) = inf{H(x, satisfies
(HO)-(H3).
In the usual
t, r. p) - H(y. t. r, 0): r E R, f E [0, T]}
way, if Ao, B. are given by (3.20)
iu(x, t) - U(y, t)j s (1 + I) (A& because the right-hand side is a supersolution on Ix - y/ < 1 in the form u(.r,y, where
- y/ + B,)
(3.3s)
of (3.36), etc. Now we seek another
r) = (1 + t)A(Ix - _tI: + E)‘~ + tB + D
A, B, D a 0, 0 < y s 1 and E > 0, which further u(x,y.
(3.37)
supersolution
(3.39)
satisfies
0) 2 mo(~.r - yj) ?= u(x, 0) - U(y. 0) = q(x)
- q(y)
(3.-lO)
and u(.Y, y, f) 2 (1 + T)(A,, + Bo) 2 u(,u, t) - U(x. f) if 1.~- y/ = 1. If we achieve
all these
things,
then
n(.r, t) - L;(.u. 1) < u(.T, s, t) < (1 + r)A&:’ 2 L tB + D by comparison.
(3.41)
(3.11)
On existence and uniqueness of solutions of Hamilton-Jacobi
equations
369
We leave it to the reader to verify that (3.41) holds for arbitrary B, D 2 0 if A = max((l
+ T)(A,
+ B,,),m,(l)
+ 1).
(3.43)
and then, with A now fixed by (3.43), (3.40) holds if D = K,(y)
= max{(m,((r/A)ii’))
- r: 0 G r CA}.
(3.44)
Finally, we use that ID,A(Ix - y(’ + e)y”/ G Ay/( .E(‘-‘)/~) to conclude that u is a supersolution on /x - yl < 1 provided B = K,(y, E) = a(Ay/(~(~-r)‘~))
+ m,(l).
(3.45)
But now we are done: A is fixed, D tends to zero with y and so the right-hand side of (3.42) can be made as small as desired by taking E small, then y small and, finally, f small. A lower bound is produced in the same way. Remark
7. Using a similar (but somewhat more complex) argument one can prove u - U E BUC(&!’ X [0, T]). In particular, if H(x, t, K, p) is independent of u, the above proof adapts to show that f u(x, t) - q(x) + H(x, t, 0) dr I0
is uniformly continuous in t with a modulus depending only on the usual data. The analogue for (SP) of these results is that if V(x) is defined implicitly by the equation V(x) + H(x, V(x), 0) = 0, and u is the solution of (SP), then u - U E BUC(b!‘). Existence
for (CP)
Theorem 4 and the same device which was used to establish existence for (SP) succeed here, and we need not repeat the argument. Remark
8. The existence results can be used together with remark 5 to strengthen the comparison results when R = R”. We illustrate this for (SP)-the analogous remarks hold for (CP). Let HE C(R” x W x R’) satisfy (HO)-(H3). Let u, u E C(R”) be a sub- and a supersolution of (SP) on W>”such that there is a constant K for which
lu(x) - u(y)1 + lu(x) - o(y)1 S K(l + Ix - yl) for x, y E W”. Then u G u. Indeed, let w E UC(WN) be the solution of (SP) provided by theorem 2. By remark 5, u G w and w < u, so the assertion is correct. Remark
9. The conditions (H3) and (H3)’ are not “natural” in the sense that they are not invariant under simple changes of variables. Hence, for example, they are not sensible on manifolds. To extend the scope of our results to include manifolds, we will formulate a generalization of (H3), and to this end let us consider the following properties of a metric (x, y) + d(x, y) on R’:
370
Lf. G. CRANDALL and P -L. Lto\s
(dl)
there
are constants
jL, A such that
(d2) for x, v E RV and 2 f _v. the map xd(x, y) is differentiable at z. We will denote the derivative of the map xd(x.v) at .K and the map yc/(x.J) at J’ b) d,(x, y) and ci,(x. y) respectively. The generalization of (H3) is: (H3)” there is a distance d satisfying (dl) and (d2) and a modulus mH such that for all (t, r) E [0, T] x 2, x, y E Rv and t 2 0 H(x, t, r, Ed,(x.y))
- H(y,
I, r. -;‘d,(.r.y))
c -mH($
- !I f
I -,L,).
The uniqueness arguments go through if (H3) is replaced by (H3)” when one replaces !.Y- ,V by d(,r,y) at appropriate points in the proofs. Moreover, one can work more generally on manifolds. The extension of the existence assertions to manifolds deserves more than a remark and will be considered elsewhere. ;IcknoMledgemenrs-The
paper. Ons of the authors and partially
supported
authors
are grateful
to H. Ishii for pointing
(MGC) was sponsored by the United by the National Science Foundation
out a shortcoming
States Army under Grant
in a previous
under Contract 60. So. &lCS-SOO3946.
draft
of this
D.A.AG79-SO-C-OOJl
REFERENCES 1. BARLES G.. Controle impulsionnel deterministe, inequations quasivariattonelles et equations de Hamtlton-Jacob] du premier ordre. these de 3’ cycle. Lniversite de Paris-IX-Dauphine (1983). 2. CRANDALL M. G., EVASS L. C. Sr LIONS P. L., Some properties of viscosity solutions of Hamtlton-Jacobi equations. Trans. Am. marh. Sot. 282. 487-502 (1984). 3. CRANDALL ICI. G. & LIOSS P. L.. Viscosity solutions of Hamilton-Jacobi equattons. Trruns. Am. ntarir. Sot. 277. l-42 (1983). 4. CRANDALL .LJ. G. & Lro,us P. L., Solutions de viscosite non bornees des equations de Hamilton-Jacobi du premtcr ordre. C. r. hebd. st%v~. Acad. Sci. Paris 298, 217-220 (1981). 5. CRANDALL hf. G. & LIONS P. L.. submitted for publication. 6. CRANDALL hf.G. & NEWCOMB R.. Viscosity solutions of Hamiiton-Jacobi equations at the boundary, Proc. Am. marh. SOC. 94, 283-290 (1985). 7. CRANDALL M. G. 8r SO~KANDIS P. E., Developments in the theory of nonlinear first order partial differential Alabama (1983). equations. Proceedings ofrhe Ituernational Symposium on Differential Equations, Birmingham. North-Holland Mathematics Studies 92, .Uorth-Holland. Amsterdam (1984). 8. ISHIIH., Remarks on the existence of viscosity solutions of Hamilton-Jacobi equations. Buil. Fat. Sci. Engng. Chuo liniu. 26, j-21 (1983). Y. JSHII H., Uniqueness of unbounded solutions of Hamilton-Jacobi equations. Indiana Uniu. .Warh. J. 33, 731-718 (1984). 10. LIONS P. L.. Generalized solutions of Hamilton-Jacobi equations. Research !Vofes in ,Lfarhemarics 69. Pitman, Boston (1982). 11. LIONS P. L., Existence results for first-order Hamilton-Jacobi equations. Ric. .Cfat. ,\‘npoll 32. 1-23 (1983). 12. SOUGANIDIS P. E., Existence of viscosity solutions of Hamilton-Jacobi equations, J. dift Eqns (to appear).