Estimates for generalized subsolutions of first order Hamilton-Jacobi equations and rearrangements

Nonlinear Analysis, Theory, Printed in Great Britain.

Methods

& Applicalrons,

Vol.

18. No.

I, pp. 9-16,

1992 0

0362-546X192 165.00+ SM 1991 Pergan0n Press plc

ESTIMATES FOR GENERALIZED SUBSOLUTIONS OF FIRST ORDER HAMILTON-JACOBI EQUATIONS AND REARRANGEMENTS ESTERGIARRUSSO Istituto di Matematica, Facolta di Scienze Matematiche, Fisiche e Naturali, UniversiG di Salerno, 84081 Baronissi (SA), Italy (Received

20 September

Key words and phrases:

1990; received

for publication 13 March 1991)

Hamilton-Jacobi

equations, rearrangements.

1. INTRODUCTION THIS WORK

concerns the Dirichlet problem for the first order Hamilton-Jacobi

equation

a.e. in Q; u = 0 on asz

Lu = 1vu1 + c(x)24 = f(x)

(1.1)

where Vu = (uX,, . . . . u,,), and ~2 is an open subset of IR”whose measure is finite. Hamilton-Jacobi equations have been studied by many authors, who investigated the existence, the uniqueness, and the stability of the solutions [7, 8, 191. Herein we employ the properties of the rearrangement of functions and isoperimetric inequalities to get comparison theorems in Lp with p = 1 or 00. In fact, denoted by U(X) a nonnegative generalized subsolution of (1 .l), by C2*the ball of lR” centered at the origin and with the same measure as Q and by (c+), and (cm)* respectively, the spherically symmetric increasing and decreasing rearrangement of the functions c+(x) = max(c(x), 0) and c-(x) = max( -c(x), 01, we consider the symmetrized operator L*w = /VW1 + ((c+)* -

and first we show that

w E wy(a*),

(c-)*)w,

Ibll, 5 I141m*

where w(x) is the decreasing spherically symmetric solution of L*w = g(x)

a.e. in CP,

w = 0 on X2*.

Above g(x) = g( Ix]) is a spherically symmetric rearrangement of f(x), which depends also on the distribution functions of c+(x) and c-(x). In particular cases (e.g. if c(x) L 0), g(x) = f*(x) (see theorem 3.1 and following remarks). Moreover, if c(x) = A = const., we show that and q(x) and o(x) being the decreasing spherically symmetric solutions respectively of L*q = IVql + Aq = f*(x)

a.e. in sZ*,

and L*w = h(x)

a.e. in a*,

w E w;*‘(a*); 9

q E w~J

10

E. GLUWJSSO

here h(x) = h(lx() is a spherically symmetric rearrangement of f(x) which depends on A. In particular cases, h(x) is equal to f*(x), e.g. if 1 I 0 (see theorem 3.2 and following remarks). This paper is understood to be a continuation of the previous ones [14, 151where the case c(x) equal to a nonpositive constant was considered. Schwartz symmetrization has been employed lately to investigate second order linear and nonlinear elliptic and parabolic problems (see for example [l-5, 10, 11, 13, 16, 23-271). 2. KNOWN

FACTS ABOUT THE REARRANGEMENTS

OF A FUNCTION

Let u(x), x E Sz C_R”, be a real valued measurable function and let p(t) = 1(x E Sz : IuI > t)l distribution function. A rearrangement of U(X) is a function g(y), y E IV, equidistributed with u(x), then if U(X)E Lp(sZ), any rearrangement of U(X)has the same LP-norm as u(x). As known, the right-continuous function

be its

U*(s) = suplt 1 0 : p(t) > s], is the decreasing rearrangement

SZO

of u(x), and U*(x) = u*(c,lxl”),

XE rn”

is the spherically symmetric decreasing rearrangement Here and below

of u(x).

n/2 en=

=

r(1 + n/2)

is the measure of the unit n-dimensional ball. Furthermore U*(S) = u*(IQzJ - s) and are the

u*(x) =

~*WnlXl”)

increasing and spherically symmetric increasing rearrangements

of u(x), respectively.

THEOREM 2.1 (cf. [17]). If u(x) and v(x) are measurable, then

s THEOREM

2.2 (cf. [22], Polya-Szego

principle). If u E Wd*p(CQand p 2 1, then

This theorem implies the absolute continuity of u*(s) in [a, Ial], with (I > 0, when U E w;*“(n), p 2 1. More details and proofs on this topic can be found in [4, 6, 18, 20, 21, 24, 261. Now let (B,], 0 5 t < ao, be a family of measurable subsets of iR, such that I&( = T and B,, c B,* if r1 < r,, then set d(a) = u*(r)

where r = inf {r 2 0 : CTE B,),

(2.1)

First order Hamilton-Jacobi

equations

11

we have (cf. [21, lemma 12, p. 511) (a)* = u* (i.e. C(a) is a rearrangement

of u(x)) and (fi > t) G B, c (ti 2 t)

if p(t) = I@: U*(s) > t)I I t I I@: U*(s) I t]J = p(t-) and u*(s) is continuous in T. Now let U(X) and Q(s), 0 I s 5 IQ/, be measurable functions and let (B,J, T 2 0, be an increasing family of subsets of IT? such that IB,I = ‘5 and B, = (s I 0 : IQ(s)] > t) if r = I@: I@(s)/ > t)J. Then denoted by G&s) the rearrangement of U(X) related to such (B,) according to (2.1) it results ([21, theorem 2, p. 521) @*(s)@(s))* = (a,)*@*(s) = U*(s)@*(s) hence

InI k.(s)

s0

IQ(4I d.s=

1621

u*(s)@*(s) ds.

(2.2)

i0

Now consider the family ID,), s E [0, IQ/], of subsets of IF? such that ID,/ = s, Ds, c Ds2 if s, < s, and D, = (x: IuJ > t) if s = p(t). Then (see [3]), if f(x) 2 0 belongs to P(Q), p 2 1, there exists a function f(it) E P(0, Isz() such that

and the following lemma holds. LEMMA

2.1 [3]. There exists a sequence of functions (f&))

that lim

Ifll fk(M~)

d.s =

s0

Idl AM@

50

all equidistributed

withf(x)

such

d.9

where g(s) E Lp”p-l’(O, l&2()if p > 1 and g(s) is a function of bounded variation on [0, JsZl]if p = 1. 3. COMPARISON

THEOREMS

Suppose U(X)E W~9’(a) verifies the inequality Lu = (VUJ + c(x)24 5 f(x),

U(X) 2 0,

a.e. in M

(3.1)

where c(x) E L”(Q)

and

f(x) EL’(Q).

(3.2)

Set c+(x) = max(c(x), 0),

c-(x) = max(-c(x),

and L*w = lvwl + ((c+)*(x) - (c-)*(x)]w the following theorems hold.

0)

E. GIARRUSSO

12

THEOREM3.1. Let c(x) and f(x) be in P(sZ), p > n, then

where w(x) is the decreasing spherically symmetric solution of the problem L*w = &,(C,[x\“)

here./&)

is the rearrangement

a.e. in a*,

W E w;*‘(n*),

of f(x) according to the function [see (2.1) and (2.2)]

Q(t) = s

exp( 1’ [-(c+),(r)

+ (c-)*(r)] g

0 tE

(3.3)

n

dr)

1% Dll.

Suppose now c(x) = A E R and consider the problems L*q = (Vql + Iq = f*(x)

a.e.

q E Wy(n*>

in Q*,

(3.4)

and L*o

=

(Vwl + A0 = &(C,lx(“)

& being the rearrangement

a.e.

in 0,

w E Woi”(Q2*),

(3.5)

of f(x) according to the function [see (2.1) and (2.2)]

then we have the following theorem. THEOREM 3.2.

If there exists a decreasing spherically symmetric solution q(x)[o(x)l [(3.5)] it results that IIAl1 5 llc?ll1

of (3.4)

[llu*(x) e-xix’ll, 5 (lo(x) e-X’x’\ll]. To prove theorem comparison lemma.

3.1 and theorem

3.2 we have first to show the following pointwise

LEMMA 3.1. Let y(s), E+(s), and c”-(s) be the functions respectively, according to lemma 2.1, then set

related to If(x)\,

c+(x),

and c-(x),

IQI

Y(s) = exp

s it results that

for a.e. s E 10, \!A(].

[F+(t) - F-(t)] g

n

,,> ,

Ial 1 u*(s) 5 u(s) = f”(t)‘I’(t) s ‘y(s) s s

0 < s I IQj,

dt

(3.6)

First order

Hamilton-Jacobi

Proof. To get (3.6) we prove that U*(S) verifies the following -1+1/n

s&l*= -u*‘(s)

+ {E+(s)-

13

equations

differential

inequality

-1+1/n

e(s))u*(s)~n

I

f_(S)Sc._ncy”

a.e. s E IO, IQ(].

’

(3.7)

a.e. s E 10, [al].) (Note that iv = J$)(s- l+“n/nC~‘n) For this purpose integrate both sides of (3.1) over the set (x: t < u I t + hJ, h > 0; t L 0, to obtain 1 ’

1

lvul dx + ;

h ,t
C+UdXIt
h

1

c-u&u + h

t
*
Then let h + 0 to get for a.e. t 2 0 --

d

because

Above P is the perimeter Moreover t

5 --

u>t

ll>t

On the other hand, formula [ 121

i

c+udx

lvu] dx - $

dt

c+(x) dx

d dt i u>t

of the De Giorgi isoperimetric

in the sense of De Giorgi

.i t
t
theorem

d dt

by definition

= I(-;

c+(x)Uti

c+(x) dx .it
ju>tc+o~)

U>f

of E+(x). Analogously, --

by definition

of c-(x)

and f(x)

d dt 1 U>f

c-W

dx = u*(P(t))e-(P(t))(-Pu’(t)),

as well as --

d dt .r

Combining

the above

15

formulas

u>t If(x)1 d-x = “mt))(-p’(t)).

we derive from (3.8)

(p(t))-‘“‘”

~[-~‘WO) + c”-(P(t))lu*wo) + _fMWJ

nCl,”

(3.8)

[9] and a Fleming-Rishel

and so --

d.Y.

(see 19, 241).

t+h c+(x)24dx 5 h

5 ;

d - tl>t If( dt .i

c-udx

(-P’(O)

n

for a.e. t E [0, ess. sup u[,

(h > 0)

E. GIARRUSSO

14

which integrated between t, and tz, with 0 I t, < t2 < ess. sup u, gives r-1+1/n P(tl) tz -

{(-E+(T)

tl 5

+

5PU2)

c"-(T))u*(T) + f(it)j___ nC;/”

dT*

From the above, by the definition of u*(s) we obtain, with h > 0, u*(s) - u*(s + h) h

5-

1 h

s

*-1+1/n

s+h

((-C+(T)

+

c”-(T))u*(T)

+ f(7)]

-iKdty G

S

which implies (3.7). Now multiply both sides of (3.7) by Y(s) to obtain (3.6), since u*(s)Y(s) is an absolutely continuous function in [a, (Szl], with a > 0. Proof of theorem 3.1. Let s -+ O+ in (3.6) to get f-l+l/n

(-E+(r) + c”-(r))---

___ ncy

dt.

Then by lemma 2.1 and theorem 2.1, lOI y(t)@(t) dt 5

JIUJJ,= u*(o+) 5

s0

InI .&(t)@(t) dt,

(3.9)

0

again by lemma 2.1 and theorem 2.1, and by (2.2). Now set w(x)

=

c’ ,X,n (-d(r) ”

+

(c-)*(r))

routine calculations show that w(x) decreases with respect to 1x1. In fact, denoted by E = (x E S2 : c(x) < 01, and by 0, , the ball of R” centered at the origin and with radius w(x) is the product of two positive and decreasing spherically symmetric functions (]E ]KJ”“, in Szg and, in a* - n*,, w(x) decreases with respect to (xl, since f*(t) is decreasing in [IEI, (&?I],because Q(t) is. [Note that if (El s t I IQl,&(t) = f*(v*(@(t))), v* being the distribution function of Q(t).] Hence w(x) verifies (3.3) and so (3.9) implies the thesis. Remark 3.1. If sup(c_)*s””

_i (n - I)Ci’”

E

(in particular if c(x) 1 0) Q(s) decreases in IO, IQ]] and so.&,(s) = f*(s), (cp. with [14, 151 for the case c(x) = const. 5 0). Remark 3.2. If n = 1 Q(t) = + exp 3 (i

t (-(c+),(r) 0

+ (c-)*(r)) dr

, >

thus j&s) increases in [0, (El] and decreases in []SJj - ]G(, IQ]], where G = (x:c(x) > 0). Hence in particular if c(x) I 0, &s) = f*(s) and if c(x) L 0, _&(s) = f*(s) (see [14, 151).

First order Hamilton-Jacobi

equations

15

Proof of theorem 3.2. By lemma 3.1 we derive

(3.10) On the other hand the function a(t) = e

-X(t/C,)““f-l+l/n

f ex(s/c,)“”

&

i 0

strictly increases in [0, +a~). This is obvious for A = 0. For 2 # 0, computing d(t) check that v t E IO, +oo[ u’(t) > 0 iff -n

1

+

k(a) = ____ n --a

cl@”

I‘

ex(s/c”)“”

it is easy to

&

0

s C”d

A

ev”/w”

n

&

+

Cna”

,h

>

0,

va

E

IO,

+a[.

(3.11)

o

Now,

1

C,O"

k’(a) = C,,an-’ ehn - n

ex(s/c,)“” &

9

90 hence, if n = 1 or if L < 0 and n E IV, k(a) strictly increases in [0, +oo) and this implies (3.1 I), since k(0) = 0. If,I>Oandn>l,wehave k”(a) = C,(n - l)P2eX”, and this implies that k’(a) > k’(0) = 0 V a > 0, so, as before, k(a) strictly increases in [0, +co). Thus, (3.11) holds v Iz E R. Hence, by (3.10), lemma 2.1, theorem 2.1 and (2.2) we obtain llnllr 5 ~"lfto4r)dt = 11~11~. 0

Analogously,

by (3.6),

11 u*(x) e-xixi 11 1

=

IRI

i0

f&MWdt

[by lemma 2.1, theorem 2.1, and by (2.2)]

Ilw(x) e-xixill, .

Remark 3.3. Of course, existence assumption

A”jQj I C, and&(t)

of theorem 3.2 is verified if L 5 0 or if ,I. > 0, e-x(t’cn)“” decreases in [0, la/].

E. GIARRUSSO

16

Remark

3.4. If I I 0 of course f^a(s) = f,(s). (In [15] it was proved also that if (n - l)/(A[ < (IQj/C,)““, then u*(x) I w(x) in Q* - Szz, L-2:being the ball centered at the origin with radius (n - l)/lA(.) If A > 0, P(t) strictly increases in [0, C,/A”] and strictly decreases in [C,/A”, (Szl] and so the vp being the distribution function of P(t). same is true for &p(t) = f*(v@(t))), Of course, j&s) = f,(s) if [sZ( 5 C,/A”. REFERENCES LIONS P. L. & TROMBETTI G.. On ontimization problems with prescribed rearrangement, Nonlinear _ (1989). results for elliptic and parabolic equations via Schwartz 2. ALVINO A., LIONS P. L. & TROMBETTI G., Comparison symmetrization, Ann. Zst. Henri Poincare 7, 37-66 (1990). per una classe di equazioni ellittiche degeneri, 3. ALVINO A. & TROMBETTI G., Sulle migliori costanti di maggiorazione 1. ALVINO A..

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