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diff --git a/macros/plain/contrib/jns/jns.dem b/macros/plain/contrib/jns/jns.dem new file mode 100644 index 0000000000..a989acf3a8 --- /dev/null +++ b/macros/plain/contrib/jns/jns.dem @@ -0,0 +1,331 @@ +% 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 |