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\maintitle{Haupttitel 14pt halbfett/Title boldface -- 14/16}
\mainrunning{Expression of Cellular Oncogenes}
\subtitle{Untertitel 10pt halbfett/Subtitle boldface -- 10/11}
\author{Ivar Ekeland@{1} and Roger Temam@{2}}
\authorrunning{R. M\"uller}
\address{@1Princeton University, Princeton NJ 08544, USA
@2Universit\'e de Paris-Sud,
Laboratoire d'Analyse Num\'erique, B\^atiment 425,\newline
F-91405 Orsay Cedex, France}
\received{June 5, 1989}
\summary{A new variant of the multi-grid algorithms is presented. It
uses multiple coarse-grid corrections with particularly associated
prolongations and restrictions. In this paper the robustness with
respect to anisotropic problems is considered.}
\keywords{multi-grid method -- coarse--grid correction --
singular perturbation -- robustness.}
\titlea{1.}{The Anisotropic Equation and Standard Multi-Grid Methods}
\titleb{1.1.}{Introduction}
Multi-grid methods are known as very fast solvers of a large class of
discretised partial differential equations. However, the multi-grid
method cannot be understood as a fixed algorithm. Usually, the
components of the multi-grid iteration have to be adapted to the given
problem and sometimes the problems are modified in order to make them
acceptable for multi-grid methods. In particular, the smoothing
iteration is the most delicated part of the multi-grid process.
An iteration is called a {\it robust} one, if it works for a sufficient
large class of problems. Attempts have been made to construct robust
multi-grid iterations by means of sophisticated smoothing processes\dots
\vfil\eject
With this chapter, the preliminaries are over, and we begin the search
for periodic solutions to Hamiltonian systems. All this will be done in
the convex case; that is, we shall study the boundary-value problem
$$\eqalign{\dot x &= JH' (t,x)\cr x(0) &= x(T)\cr}$$
with $H(t,\bullet )$ a convex function of $x$, going to $+\un$ when
$\lss x\rss \to \un$.
\titleb{1.2.}{Autonomous Systems}
In this section, we will consider the case when the Hamiltonian $H(x)$
is autonomous. For the sake of simplicity, we shall also assume that it
is $C^1$.
We shall first consider the question of nontriviality, within the
general framework of $\lr \Aun , \Bun\rr$-subquadratic Hamiltonians. In
the second subsection, we shall look into the special case when $H$ is
$\lr 0,\bun\rr$-subquadratic, and we shall try to derive additional
information.
\titlec{ The General Case: Nontriviality.}
We assume that $H$ is $\lr \Aun , \Bun \rr$-sub\-qua\-dra\-tic at infinity,
for some constant symmetric matrices $\Aun$ and $\Bun$, with $\Bun
-\Aun$ positive definite. Set:
$$\eqalignno{
\gamma :& = {\rm smallest\ eigenvalue\ of}\ \ \Bun - \Aun & (1)\cr
\lambda : & = {\rm largest\ negative\ eigenvalue\ of}\ \ J {d\ov dt} +\Aun\
. & (2)\cr}$$
Theorem 21 tells us that if $\lambda +\gamma < 0$, the boundary-value
problem:
$$\eqalign{ \dx &= JH' (x)\cr
x(0) &= x (T)\cr}\eqno(3)$$
has at least one solution $\ol x$, which is found by minimizing the dual
action functional:
$$ \psi (u) = \int_o^T \lb \12 \lr \Lai_o^{-1} u,u\rr + N^\ast (-u)\rb
dt\eqno(4)$$
on the range of $\Lai$, which is a subspace $R (\Lai )\sb L^2$ with
finite codimension. Here
$$ N(x) := H(x) - \12 \lr \Aun x,x\rr\eqno(5)$$
is a convex function, and
$$ N(x) \le \12 \lr \lr \Bun - \Aun\rr x,x\rr + c\ \ \ \fa x\
.\eqno(6)$$
\proposition{ 1.} { Assume $H'(0)=0$ and $ H(0)=0$. Set:
$$ \de := \liminfu_{x\to 0} 2 N (x) \lss x\rss^{-2}\ .\eqno(7)$$
If $\gamma < - \lambda < \de$, the solution $\ol u$ is non-zero:
$$ \ol x (t) \ne 0\ \ \ \fa t\ .\eqno(8)$$}
\proof{} Condition (7) means that, for every $\de ' > \de$, there is
some $\ep > 0$ such that
$$ \lss x\rss \le \ep \Rightarrow N (x) \le {\de '\ov 2} \lss x\rss^2\
.\eqno(9)$$
It is an exercise in convex analysis, into which we shall not go, to
show that this implies that there is an $\eta > 0$ such that
$$ f\lss x\rss \le \eta \Rightarrow N^\ast (y) \le {1\ov 2\de '} \lss
y\rss^2\ .\eqno(10)$$
\begfig 1.5cm
\figure{1}{This is the caption of the figure displaying a white eagle
and a white horse on a snow field}
\endfig
Since $u_1$ is a smooth function, we will have $\lss hu_1\rss_\un \le
\eta$ for $h$ small enough, and inequality (10) will hold, yielding
thereby:
$$ \psi (hu_1) \le {h^2\ov 2} {1\ov \lambda} \lss u_1 \rss_2^2 + {h^2\ov 2}
{1\ov \de '} \lss u_1\rss^2\ .\eqno(11)$$
If we choose $\de '$ close enough to $\de$, the quantity $\lr {1\ov \lambda}
+ {1\ov \de '}\rr$ will be negative, and we end up with
$$ \psi (hu_1) < 0\ \ \ \ \ \for\ \ h\ne 0\ \ {\rm small}\ .\eqno(12)$$
On the other hand, we check directly that $\psi (0) = 0$. This shows
that 0 cannot be a minimizer of $\psi$, not even a local one. So $\ol u
\ne 0$ and $\ol u \ne \Lai_o^{-1} (0) = 0$. \qed
\corollary{ 2.} { Assume $H$ is $C^2$ and $\lr \aun
,\bun\rr$-subquadratic at infinity. Let
$\xi_1,\allowbreak\dots,\allowbreak\xi_N$ be the
equilibria, that is, the solutions of $H' (\xi ) = 0$. Denote by $\om_k$
the smallest eigenvalue of $H'' \lr \xi_k\rr$, and set:
$$ \om : = \Min \lg \om_1 , \dots , \om_k\rg\ .\eqno(13)$$
If:
$$ {T\ov 2\pi} \bun < - E \lb - {T\ov 2\pi}\aun\rb < {T\ov
2\pi}\om\eqno(14)$$
then minimization of $\psi$ yields a non-constant $T$-periodic solution
$\ol x$.}
We recall once more that by the integer part $E [\al ]$ of $\al \in
\RR$, we mean the $a\in \ZZ$ such that $a< \al \le a+1$. For instance,
if we take $\aun = 0$, Corollary 2 tells us that $\ol x$ exists and is
non-constant provided that:
$$ {T\ov 2\pi} \bun < 1 < {T\ov 2\pi}\eqno(15)$$
or
$$ T\in \lr {2\pi\ov \om},{2\pi\ov \bun}\rr\ .\eqno(16)$$
\proof{} The spectrum of $\Lai$ is ${2\pi\ov T} \ZZ +\aun$. The
largest negative eigenvalue $\lambda$ is given by ${2\pi\ov T}k_o +\aun$,
where
$$ {2\pi\ov T}k_o + \aun < 0\le {2\pi\ov T} (k_o +1) + \aun\
.\eqno(17)$$
Hence:
$$ k_o = E \lb - {T\ov 2\pi} \aun\rb \ .\eqno(18)$$
The condition $\gamma < -\lambda < \de$ now becomes:
$$ \bun - \aun < - {2\pi\ov T} k_o -\aun < \om -\aun\eqno(19)$$
which is precisely condition (14).\qed
\lemma {3.} { Assume that $H$ is $C^2$ on $\RRn \sm \{ 0\}$ and
that $H'' (x)$ is non-degenerate for any $x\ne 0$. Then any local
minimizer $\tx$ of $\psi$ has minimal period $T$.}
\proof{} We know that $\tx$, or $\tx + \xi$ for some constant $\xi
\in \RRn$, is a $T$-periodic solution of the Hamiltonian system:
$$ \dx = JH' (x)\ .\eqno(20)$$
There is no loss of generality in taking $\xi = 0$. So $\psi (x) \ge
\psi (\tx )$ for all $\tx$ in some neighbourhood of $x$ in $W^{1,2} \lr
\RR / T\ZZ ; \RRn\rr$.
But this index is precisely the index $i_T (\tx )$ of the $T$-periodic
solution $\tx$ over the interval $(0,T)$, as defined in Sect.~2.6. So
$$ i_T (\tx ) = 0\ .\eqno(21)$$
Now if $\tx$ has a lower period, $T/k$ say, we would have, by Corollary
31:
$$ i_T (\tx ) = i_{kT/k}(\tx ) \ge ki_{T/k} (\tx ) + k-1 \ge k-1 \ge
1\ .\eqno(22)$$
This would contradict (21), and thus cannot happen.\qed
\titled{Notes and Comments.} The results in this section are a
refined
version of [CE1]; the minimality result of Proposition 14 was the first
of its kind.
To understand the nontriviality conditions, such as the one in formula
(16), one may think of a one-parameter family $x_T$, $T\in \lr
2\pi\om^{-1}, 2\pi \bun^{-1}\rr$ of periodic solutions, $x_T (0) = x_T
(T)$, with $x_T$ going away to infinity when $T\to 2\pi \om^{-1}$, which
is the period of the linearized system at 0.
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\theorem{4 (Ghoussoub-Preiss).} { Assume $H(t,x)$ is
$(0,\ep )$-subquadratic at
infinity for all $\ep > 0$, and $T$-periodic in $t$
$$ H (t,\bullet )\ \ \ \ \ {\rm is\ convex}\ \ \fa t\eqno(23)$$
$$ H (\bullet ,x)\ \ \ \ \ {\rm is}\ \ T{\rm -periodic}\ \ \fa x
\eqno(24)$$
$$ H (t,x)\ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with}\ \ n (s)s^{-1}\to
\un\ \ {\rm as}\ \ s\to \un\eqno(25)$$
$$ \fa \ep > 0\ ,\ \ \ \exists c\ :\ H(t,x) \le {\ep\ov 2}\lss x\rss^2 +
c\ .\eqno(26)$$
Assume also that $H$ is $C^2$, and $H'' (t,x)$ is positive definite
everywhere. Then there is a sequence $x_k$, $k\in \NN$, of $kT$-periodic
solutions of the system
$$ \dx = JH' (t,x)\eqno(27)$$
such that, for every $k\in \NN$, there is some $p_o\in\NN$ with:
$$ p\ge p_o\Rightarrow x_{pk} \ne x_k\ .\eqno(28)$$\qed}
\example {1 {\rm(External forcing).}}{ Consider the system:
$$ \dx = JH' (x) + f(t)\eqno(29)$$
where the Hamiltonian $H$ is $\lr 0,\bun\rr$-subquadratic, and the
forcing term is a distribution on the circle:
$$ f = {d\ov dt} F + f_o\ \ \ \ \ {\rm with}\ \ F\in L^2 \lr \RR / T\ZZ
; \RRn\rr\ ,\eqno(30)$$
where $f_o : = T^{-1}\int_o^T f (t) dt$. For instance,
$$ f (t) = \sum_{k\in \NN} \de_k \xi\ ,\eqno(31)$$
where $\de_k$ is the Dirac mass at $t= k$ and $\xi \in \RRn$ is a
constant, fits the prescription. This means that the system $\dx = JH'
(x)$ is being excited by a series of identical shocks at interval $T$.}
\definition{5.}{Let $A_\un (t)$ and $B_\un (t)$ be symmetric
operators in $\RRn$, depending continuously on $t\in [0,T]$, such that
$A_\un (t) \le B_\un (t)$ for all $t$.
A Borelian function $H: [0,T]\times \RRn \to \RR$ is called $\lr A_\un
,B_\un\rr$-{\it subquadratic at infinity} if there exists a function
$N(t,x)$ such that:
$$ H (t,x) = \12 \lr A_\un (t) x,x\rr + N(t,x)\eqno(32)$$
$$ \fa t\ ,\ \ \ N(t,x)\ \ \ \ \ {\rm is\ convex\ with\ respect\ to}\
\ x\eqno(33)$$
$$ N(t,x) \ge n\lr \lss x\rss\rr\ \ \ \ \ {\rm with}\ \ n(s)s^{-1}\to
+\un\ \ {\rm as}\ \ s\to +\un\eqno(34)$$
$$ \exists c\in \RR\ :\ \ \ H (t,x) \le \12 \lr B_\un (t) x,x\rr + c\ \
\ \fa x\ .\eqno(35)$$
}
If $A_\un (t) = a_\un I$ and $B_\un (t) = b_\un I$, with $a_\un \le
b_\un \in \RR$, we shall say that $H$ is $\lr a_\un
,b_\un\rr$-subquadratic at infinity. As an example, the function $\lss x
\rss^\al$, with $1\le \al < 2$, is $(0,\ep )$-subquadratic at infinity
for every $\ep > 0$. Similarly, the Hamiltonian
$$ H (t,x) = \12 k \lss k\rss^2 +\lss x\rss^\al\eqno(36)$$
is $(k,k+\ep )$-subquadratic for every $\ep > 0$. Note that, if $k<0$,
it is not convex.
\titled{Notes and Comments.} The first results on subharmonics were
obtained by Rabinowitz in [Ra1], who showed the existence of infinitely
many subharmonics both in the subquadratic and superquadratic case, with
suitable growth conditions on $H'$. Again the duality approach enabled
Clarke and Ekeland in [CE2] to treat the same problem in the
convex-subquadratic case, with growth conditions on $H$ only.
Recently, Michalek and Tarantello (see [MT1] and [Ta1]) have obtained
lower bound on the number of subharmonics of period $kT$, based on
symmetry considerations and on pinching estimates, as in Sect.~5.2 of
this article.
\begref{References}{[MT1]}
\refmark{[CE1]} Clarke, F., Ekeland, I.: Nonlinear oscillations and
boundary-value problems for Hamiltonian systems. Arch. Rat. Mech. Anal.
{\bf 78} (1982) 315--333
\refmark{[CE2]} Clarke, F., Ekeland, I.: Solutions p\'eriodiques, du
p\'eriode donn\'ee, des \'equations hamiltoiennes. Note CRAS Paris {\bf
287} (1978) 1013--1015
\refmark{[MT1]} Michalek, R., Tarantello, G.: Subharmonic solutions with
prescribed minimal period for nonautonomous Hamiltonian systems. J.
Diff. Eq. {\bf 72} (1988) 28--55
\refmark{[Ta1]} Tarantello, G.: Subharmonic solutions for Hamiltonian
systems via a $\bbbz_p$ pseudoindex theory. Annali di Mathematica Pura
(to appear)
\refmark{[Ra1]} Rabinowitz, P.: On subharmonic solutions of a Hamiltonian
system. Comm. Pure Appl. Math. {\bf 33} (1980) 609--633
\endref
\byebye