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+% This is JNS.DEM the demonstration file of
+% the plain TeX macro package from Springer-Verlag
+% for the Journal of Nonlinear Science
+% version of 16 July 1990
+\def\12{{1\ov 2}}
+\def\al{\alpha}
+\def\Aun{A_\un}
+\def\aun{a_\un}
+\def\bullet{\cdot}
+\def\Bun{B_\un}
+\def\bun{b_\un}
+\def\de{\delta}
+\def\dx{\dot x}
+\def\ep{\epsilon}
+\def\fa{\forall}
+\def\for{{\rm for}}
+\def\Lai{\Lambda}
+\def\lb{\left[}
+\def\lg{\left\{}
+\def\degr{\hbox{$^\circ$}}
+\def\arcmin{\hbox{$^\prime$}}
+\def\arcsec{\hbox{$^{\prime\prime}$}}
+\def\liminfuu{{\rm lim inf}$\,$}
+\def\liminfu{\mathop{\vphantom{\tst\sum}\hbox{\liminfuu}}}
+\def\limsupuu{{\rm lim sup}$\,$}
+\def\limsupu{\mathop{\vphantom{\tst\sum}\hbox{\limsupuu}}}
+\def\lr{\left(}
+\def\lss{\left\|}
+\def\Min{{\rm Min\,}}
+\def\NN{\bbbn}
+\def\ol{\overline}
+\def\om{\omega}
+\def\ov{\over}
+\def\rb{\right]}
+\def\rg{\right\}}
+\def\RRn{\bbbr^{2n}}
+\def\RR{\bbbr}
+\def\rr{\right)}
+\def\rss{\right\|}
+\def\sm{\setminus}
+\def\tst{\textstyle}
+\def\tx{\wt x}
+\def\un{\infty}
+\def\wt{\widetilde}
+\def\ZZ{\bbbz}
+\ifx\BlankIsSpace\undefined % <| Is this file
+\input jns.cmm %<---| called by
+\else\vfill\eject\pageno=26\relax %<---|
+\def\ident{Macro Package for Authors %<---| JNS.DOC
+Coding with Plain\TeX{} -- %<---|
+Demonstration File}\let\INS=N %<---| or
+\let\speciali=\undefined\fi %<| standalone?
+%
+\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.
+\vskip8 true mm
+\tabcap{1}{Observational results from NGC 4827}
+\vbox{\petit\hrule\smallskip
+\vbox{\halign{\enspace#\hfil\enspace&&#\hfil\enspace\cr
+&&\multispan3{RA (1950)}\hfil&\ &\multispan3{Dec (1950)}\hfil
+ & $S$ & Pol & \% & $\log P$\cr
+\noalign{\vskip-2mm}
+&&\multispan3\hrulefill\quad&&\multispan3\hrulefill\quad&&\cr
+\noalign{\vskip-0.5mm}
+ & &(h) &(m) & (s) & & (\degr) & (\arcmin) & (\arcsec)
+ & (mJy) & (mJy) & & (W Hz$^{-1}$)\cr
+\noalign{\smallskip}
+\noalign{\hrule}
+\noalign{\smallskip}
+Core & (5 GHz) & 12 & 54 & 18.0 & & 27 & 26 & 56.2
+ & 8 & & & 21.64 \cr
+Total&(327 MHz)& & & & & & &
+ & 210 & & & 23.13 \cr
+ &(1.4 GHz)& & & & & & &
+ & 57 & 1.3 & 2 & 22.49 \cr
+ & (5 GHz) & & & & & & &
+ & 26 & 0.73 & 3 & 22.15 \cr}
+\smallskip\hrule}}
+\vskip 8 true mm
+
+\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