Hyperbolicity: Lectures given at the Centro Internazionale Matematico Estivo (C.I.M.E.), held in Cortona (Arezzo), Italy, June 24 – July 2, 1976

234 Pages·2011·9.227 MB·English-French

by  Haim Brezis (auth.), Giuseppe Da Prato, Giuseppe Geymonat (eds.)

Giuseppe Da Prato.Giuseppe Geymonat(Eds.)
Hyperbolicity
Lecturesgiven atthe
Centro Internazionale Matematico Estivo (C.I.M.E.),
held inCortona(Arezzo), Italy,
June 24 -July 2, 1976
FONDAZIONE
CIME
~
Springer
ROBERTO CONTI
C.LM.E. Foundation
clo Dipartimento di Matematica"U. Dini"
Viale Morgagni n.67/a
50134 Firenze
Italy
[email protected]
ISBN 978-3-642-11104-4 ISBN 978-3-642-11105-1 (eBook)
DOlI0.1007/978-3-642-11105-1
Springer Heidelberg Dordrecht London New York
©Springer-Verlag Berlin Heidelberg2011
Reprint ofthe t"ed. C.LM.E., Ed. Liguori,Napoli 1977
With kind permission ofC.LM.E.
Printed on acid-free paper
Springer.com
CENTRO INTERNATIONALE MATEMATICO ESTIVO
(C.LM.E)
II Ciclo - Cortonada124 giugno a121uglio 1976
HYPERBOLICITY
H.Brezis: Firstorder quasilinearequationson atorus ; 5
J. Chazarain-A.Piriou: Problemesmixteshyperboliques:
Premierepartie:Lesproblemesmixtes
hyperboliquesverifiantlaconditionde
Lopatinskiuniforme 17
Deuxieme partie:Propagation et reflexion des
singularites.................................................................. 43
L. Garding: Introduction to hyperbolicity...................................... 73
T.Kato: Linear andquasi-linear equationsof
evolution ofhyperbolic type 125
K.W.Morton: Numerical methodsfornon-linear hyperbolic
equationsofmathematicalphysics 193
CENTRO INTERNAZIONALE MATEMATICO ESTIVO
FIRST ORDER QUASILINEAR EQUATIONS ON A TORUS
H. BREZIS
Corso tenuto a Cortona dal 24 giugno al 21uglio 1976
FIRST ORDER QUASILINEAR EQUATIONS 6N A TORUS
Haim BREZIS
We report on a joint work with L. Nirenberg detailed proofs
are given in [2].
We consider here real functions of x:: (xl' •••,x in /Kn
n)
which are periodic of period 2rt in each variable, i.e., func-
tions defined on the torus J2... , and on.J1. we consider the
constant coefficient operator
d
A u = --u ,
Let g E: C(IR) and let t <; C(.IL) ; our purpose is to find a
real periodic function u on JL satisfying
(1) A u + g(u) :: f'{x) •
We will give necessary and sufficient conditions for a solution
to exist in case g is increasing and a sufficient condition
in the general case.
~ 2 J
Let N(A):: u E L Au:: 0 (to be under-st ood in the
2
distribution sense). Let P denote the L projec ,ion on
NCA). P has the important property that P f ~ 0 when
8
f ~ 0 • Indeed, let f). = (I +). A)-I, so that
....
the maximum principle. We shall verify that f>. P f in
L2 as ). ~ +ov • We have (A f). ,v) = 0 ~ v€, N(A) and so
(f - f). ,v) = 0 Vv e N(A). Since If.AI L'l.. ~ If I L1- we may
suppose f ;. ->. if as ,\,-t +01> and if € N(A). Finally IfA/2L1-=
1'l.
,
= (f, f ) and the conclusion follows directly.
....
Note that P 1 =1 and thus P is a contraction in LoO
Set
g+ = limin! g(u) g- =limsup g(u)
u ~ to'Q U ~_oo
Our main result is the following
Theorem 1. Assume g E C( IR.) is locally':bf bounded variation
~;>
and let f ( C(n.) • Suppose there is a 0 such that
'::' P f< g -f
, +
Then (1) has a solution
We first derive a Corollary of Theorem 1.
Corollary 2. Assume g e c(1R) is increasing and let f 6 C(..n.).
Equati on (1) has a (unique) solution ufL00 if and only if'
there is a ~> 0 such that (2) holds.
Proof of Corollary 2 from Theorem 1.
V
Observe first that since (A u u) = 0 u f D(A) , it fol-
l ows easily that N(A) = R(A)L •
9
Necessity. If ~ is a solution of (1) we have P g(u) =P f
Let M~ s~p ess u ; since g is increasing we find g(u) ~
g(M) and therefore P g(u) ~ geM) • Hence P f ~ g(M) <g
+
a similar argument yields the left hand side of (2).
Sufficiency follows from Theorem 1 (g is locally of bounded
variation since it is increasing). Uniqueness is obv io~s by
monotonicity.
The following Lemma plays an essential role in the proof of
Theorem 1.
Lemma 1. The set of functions of the form
AV+S , v ~ C00
is dense in the space of continuous functions.
Proof. If f G C we may approximate f arbitrarily close in
the maximum norm by a real finite trigonometric sum
;...; = ') ik.x
L.. c e
k
Ikl~N
Here k = (~, •••,kn) represents a multi-index of integers,
k.x = l k .x . and Ikl = z..1kJ.1 • Then we have f = A v + ~
J J
where
~I c ik.x ~ L ik.x
v = k e = c e
k
i 1<.'1 a.k k(T
Here J = [k jkl ~ N and a.k F OJ
J'= lk lxls N and a.k = °1
10
Note that since f is real valued, so are v and ?; •
Proof of Theorem 1. We first construct super and subsolutions
u
~ u , i.e., functions satisfying
A u + g(a) - f ~ 0 ~ A ~ + g(~) - f •
To this end let M be such that
g(u) ~ ~ g - ~/4 for u ~ M •
u~t OQ
By Lemma 1 we may write
r for U € C00, ?; f" C""I"1N(A),IR/ c ~ /4.
e.
By adding a large constant to u we may always suppose U M.
We have P f = P~ +PR=~+ PR and consequently f ~
~ lim g - 3·~/4. Hence A ii + g(u) - f = - ~- R + g(ii) e o.
U -'>'1"'»
Simi l arly we construct ~ which we may always take to satisfy
Now we use a monotone iteration scheme. Let gl' g2 f C(iR..) be
non decreasing, bounded functions such that g on
I
min ~ , max «] . Since g is continuous and locally of bound
ed variation, such a decomposition is possible on the finite in-
uJ
~
terval [min , max (see e.g. [3] , Theoreme 24-6 and its
Corollaires). Then we extend gl and g to be for instance
2
constant outside [min ~ , max Ii] • We will solve
using the fact that T is order preserving.
11
2•
First note that I + gl + A is invertible in L Indeed A
2
is maximal monotone, gl is monotone continuous on L and
thus A + gl is maximal monotone (see e g, [1] Corollaire
s
2.7). It is easily verified that (I + gl + A)-l is order
preserving and so is T.
Since £' u are sub and super solutions we see that £ ~ T £ ,
u
~ T u • Consequently the sequences
satisfy un ~ vn ' un1', vn J,. • Hence both sequences un'vn
converge to functiono ~, u and u and u
are solutions of (l)~
We illustrate Theorem 1 with two examples.
1 2 n
Example 1. Assume a , a , ,a are linearly independent
over the rationals (d,e., a.k = 0 with k E" Z n implies k 0).
Then (2) takes the form
I~I J
e; c f dx < g
...l1.. +
Indeed in this case N(A) is reduced to constant functions so
that P f is the average of f over.fL. All characteristics
of A are dense curves (on the torus).
Example 2. Suppose n =2 , a1 = a2 =1 • Then (2) takes the
r;>
form : there is a 0 such that ifr E [0 ~ 2rr]
.2.rr
~ -112T Jr f(N S, s) ds .<:: g+ - r
o
12
In this case N(A) consists of all functions of the form
r
2
(x ) , <P £ L • Here, characteristics of A are close1i
l-x2
curves (on the torus) and in fact (1) can be solved along the
characteristics.
Remarks.
1) The proof of Theorem 1 may be extended to the equation
A u + g(x,u) = 0
under appropriate assumptions on g(x,u).
2) If is such that g'> 0 and f satisfies (2)
then the solution u of (1) is Coo. This can be proved by
a technique of elliptic regularization. Consider u solu-
tion of
f
Using sub and super solutions we first verify that lu~l~c as
E ~ O. Then by differentiating (3) k times we obtain
lu~IC s C with the aid of the maximum principle.
K.
3) Theorem 1 can be extended to variational inequalities - or
more generally to maximal monotone graphs g (instead of
continuous g). More precisely, let g be a maximal monotone
graph in fR and set g+ = Sup R(g), g_ = 1m R(g) • Let
f E C(Q.) be such that (2f, holds. Then there exists a solu-
tion u ~ L00 of