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author | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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committer | Norbert Preining <norbert@preining.info> | 2019-09-02 13:46:59 +0900 |
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diff --git a/macros/latex209/contrib/lms/lmssampl.tex b/macros/latex209/contrib/lms/lmssampl.tex new file mode 100644 index 0000000000..f946b5d26f --- /dev/null +++ b/macros/latex209/contrib/lms/lmssampl.tex @@ -0,0 +1,630 @@ +% lmssampl.tex +% Copyright (C) 1993,1994,1995 Cambridge University Press +% v0.3, 4th January 1995 + +\documentstyle{lms} + +%%% Uncomment ONE of the combinations \onetrue, \twotrue... \seventrue, +%%% depending on the LaTeX configuration on your system. The files in +%%% parentheses are required to tex the sample pages and guide. + +%%% You will need to have a local copy of the files shown in parentheses +%%% for those documentstyles. If you have any problems at all, remember +%%% that your best option is to choose the version which has `noams' as +%%% part of its title, and if you wish the typesetter to substitute ams +%%% fonts for your fonts, please indicate this in the text. + +%%% oldfss = LaTeX without NFSS +%%% nfssone = LaTeX with New Font Selection Scheme v1 +%%% nfsstwo = LaTeX with New Font Selection Scheme v2 + +%%% noams = No AMS fonts, sample pages will substitute some characters +%%% amsone = AMS v1 fonts +%%% amstwo = AMS v2 fonts (the recommended option) + +%%% Uncomment ONE of the following combinations: + + \onetrue % oldfss/noams +% \twotrue % oldfss/amsone (mssymb.tex) +% \threetrue % oldfss/amstwo (amssym.def,amssym.tex) +% \fourtrue % nfssone/noams +% \fivetrue % nfssone/amstwo (amsfonts.sty,amssymb.sty) +% \sixtrue % nfsstwo/noams +% \seventrue % nfsstwo/amstwo (amsfonts.sty,amssymb.sty) + +%%%% DON'T MODIFY THE FOLLOWING CODE; SKIP TO `START HERE' %%%% + +\ifCUPmtlplainloaded + \ifoldfss +% \usefraktur + \usescript + % + % set up 9pt Bbb for bibliography + \font\ninemsy=mtym10 at 9pt + \def\bibliobbb{\textfont\msyfam=\ninemsy} + \fi +% +\else +% +%%% one (oldfss/noams) + \ifone + \def\frak{\protect\cal} + \let\goth=\frak + \def\Bbb{\protect\bf} + \def\scr{\protect\cal} + \fi + +%%% two (oldfss/amsone) + \iftwo + \input mssymb\relax % %%%MR own catcodes + \makeatletter + % + % if the following is already uncommented in your version of + % mssymb.tex, you may comment this out to % FINISH HERE to save + % a font family + % + % set up Euler Fraktur/Gothic font (\frak, \goth) + \font\teneuf=eufm10 + \font\seveneuf=eufm7 + \font\fiveeuf=eufm5 + \newfam\euffam + \textfont\euffam=\teneuf + \scriptfont\euffam=\seveneuf + \scriptscriptfont\euffam=\fiveeuf + \def\frak{\ifmmode\let\next\frak@\else + \def\next{\errmessage{Use \string\frak\space only in math mode}}\fi + \next} + \def\goth{\ifmmode\let\next\frak@\else + \def\next{\errmessage{Use \string\goth\space only in math mode}}\fi + \next} + \def\frak@#1{{\frak@@{#1}}} + \def\frak@@#1{\fam\euffam#1} + % FINISH HERE + % + % set up 9pt Bbb for bibliography + \font\ninemsb=msym9 + \def\bibliobbb{\textfont\msyfam=\ninemsb} + % + \def\scr{\protect\cal} + \let\le\leqslant \let\ge\geqslant + \let\leq\leqslant \let\geq\geqslant + \makeatother + \fi + +%%% three (oldfss/amstwo) + \ifthree +% \input test/oldfss/amssym.def\relax +% \input test/oldfss/amssym.tex\relax %%%MR own catcode + \input amssym.def\relax + \input amssym.tex\relax %%%MR own catcode + % + % set up Euler Script font (\scr) + \font\scrten=eusm10 \skewchar\scrten='60 + \font\scrseven=eusm7 \skewchar\scrseven='60 + \font\scrfive=eusm5 \skewchar\scrfive='60 + \newfam\scrfam + \textfont\scrfam=\scrten + \scriptfont\scrfam=\scrseven + \scriptscriptfont\scrfam=\scrfive + \def\scr{\fam\scrfam\scrten} + % + % set up 9pt Bbb for bibliography + \font\ninemsb=msbm9 + \def\bibliobbb{\textfont\msbfam=\ninemsb} + % + \let\le\leqslant \let\ge\geqslant + \let\leq\leqslant \let\geq\geqslant + \fi + +%%% four (nfssone/noams) + \iffour + \def\frak{\protect\cal} + \let\goth=\frak + \def\Bbb{\protect\mathbf} + \def\scr{\protect\cal} + \fi + +%%% five (nfssone/amstwo) + \iffive + \makeatletter +% \input test/nfss1/amsfonts.sty\relax %%%MR assumes \makeatletter +% \input test/nfss1/amssymb.sty\relax + \input amsfonts.sty\relax %%%MR assumes \makeatletter + \input amssymb.sty\relax + % + \new@fontshape{eus}{m}{n}{% + <5>eusm5<6>eusm6<7>eusm7<8>eusm8<9>eusm9<10>eusm10% + <11>eusm10 at 10.95pt<12>eusm10 at 12pt<14>eusm10 at 14.4pt% + <17>eusm10 at 17.28pt<20>eusm10 at 20.736pt% + <25>eusm10 at 24.8832pt}{}% + % + \new@fontshape{eus}{b}{n}{% + <5>eusb5<6>eusb6<7>eusb7<8>eusb8<9>eusb9<10>eusb10% + <11>eusb10 at 10.95pt<12>eusb10 at 12pt<14>eusb10 at 14.4pt% + <17>eusb10 at 17.28pt<20>eusb10 at 20.736pt% + <25>eusb10 at 24.8832pt}{}% + % + \extra@def{eus}{\skewchar#1='60}{} + % + \newmathalphabet{\scr} + \addtoversion{normal}{\scr}{eus}{m}{n} + \addtoversion{bold}{\scr}{eus}{b}{n} + % + \let\le\leqslant \let\ge\geqslant + \let\leq\leqslant \let\geq\geqslant + \makeatother + \fi + +%%% six (nfsstwo/noams) + \ifsix + \let\cal\mathcal + \def\frak{\protect\mathcal} + \let\goth=\frak + \def\Bbb{\protect\mathbf} + \def\scr{\protect\mathcal} + \fi + +%%% seven (nfsstwo/amstwo) + \ifseven + \makeatletter +% \input test/nfss2/amsfonts.sty\relax %%%MR asuumes \makeatletter +% \input test/nfss2/amssymb.sty\relax + \input amsfonts.sty\relax %%%MR asuumes \makeatletter + \input amssymb.sty\relax + % + \DeclareMathAlphabet{\scr}{U}{eus}{m}{n} + \SetMathAlphabet{\scr}{bold}{U}{eus}{b}{n} + % + \let\cal\mathcal + \let\le\leqslant \let\ge\geqslant + \let\leq\leqslant \let\geq\geqslant + \makeatother + \fi +% +\fi % end of CUPmtlplainloaded + + +%%% for guide only + +\ifoldfss + \newcommand{\mitbf}[1] {\mbox{\boldmath ${#1}$}} + \newcommand{\rmn}[1] {{\rm {#1}}} + \newcommand{\itl}[1] {{\it {#1}}} + \newcommand{\bld}[1] {{\bf {#1}}} +\fi + +\ifnfssone + \newmathalphabet{\mathit} + \addtoversion{normal}{\mathit}{cmr}{m}{it} + \addtoversion{bold}{\mathit}{cmr}{bx}{it} + \newmathalphabet{\mathcal} + \addtoversion{normal}{\mathcal}{cmsy}{m}{n} + \newcommand{\mitbf}[1] {\hbox{\mathversion{bold}${#1}$}} + \newcommand{\rmn}[1] {{\mathrm {#1}}} + \newcommand{\itl}[1] {{\mathit {#1}}} + \newcommand{\bld}[1] {{\mathbf {#1}}} +\fi + +\ifnfsstwo + \newcommand{\mitbf}[1] {\hbox{\mathversion{bold}${#1}$}} + \newcommand{\rmn}[1] {{\mathrm {#1}}} + \newcommand{\itl}[1] {{\mathit {#1}}} + \newcommand{\bld}[1] {{\mathbf {#1}}} +\fi + +%%% START HERE %%% + +\extraline{First author supported in part by a grant + from the National Science Foundation} + +\newtheorem{theorem}{Theorem} +\newtheorem{lemma}{Lemma} +\newtheorem{corollary}{Corollary} +\newunnumbered{remark}{Remark} +\newunnumbered{remarks}{Remarks} +\newunnumbered{notation}{Notation} +\newunnumbered{definition}{Definition} + +\classno{35B60} +\begin{document} + +\title[Dirichlet's Problem When The Data Is An Entire Function] + {Dirichlet's Problem When The Data\\ Is An Entire Function} + +\author{Dmitry Khavinson \and\ Harold S. Shapiro} + +\maketitle + +\section{Introduction} +\label{sec-Introduction} + +This paper may be regarded as a sequel (and correction) to \cite{Incorrect +version}, and we use similar notations. Thus $x=(x_1,\ldots,x_n)$ and +$y=(y_1,\dots,y_n)$ denote points of~${\Bbb R}^n$ and $z=x+iy$ a point +of~${\Bbb C}^n$. We use standard multi-index notations; thus for +$\alpha=(\alpha_1,\dots,\alpha_n)$ with $\alpha_j$ non-negative integers, +$z^\alpha=z_1^{\alpha_1}\!\dots z_n^{\alpha_n}$, +$|\alpha|=\alpha_1+\dots+\alpha_n$, $\alpha!=\alpha_1!\dots\alpha_n!$ and +$|z|=(|z_1|^2+\dots|z_n|^2)^{1/2}$. + +$D$ denotes $(D_1,\dots,D_n)$ with $D_j=\partial/\partial z_j$, and +$\partial$~denotes $(\partial_1,\ldots,\partial_n)$ with +$\partial_j=\partial/\partial x_j$ (or $\partial/\partial y_j$, +etc., as the case may be). + +${\scr P}_{m,n}$ denotes the set of polynomials on $n$~letters with complex +coefficients, of degree at most~$m$, and ${\scr H}_{m,n}$ the set of +homogeneous polynomials of degree~$m$ in ${\scr P}_{m,n}$ augmented by~$0$ +(so that ${\scr H}_{m,n}$ is a vector space over $\Bbb C$). The number of +variables~($n$) will usually be suppressed in the notation, and we shall then +write ${\scr P}_m{\scr H}_m$. + +For $f\in{\scr P}_m$, $f^*$~is the polynomial obtained from~$f$ by +conjugating its coefficients. + +$E_n$ denotes the set of entire functions on~${\Bbb C}^n$, and $X_n$ the entire +functions of exponential type. $E_n$~and~$X_n$ may be considered as topological +vector spaces, so as to be duals of one another, in a standard way (compare +\cite{Incorrect version}). $F_n$~is the Hilbert space of entire functions~$f$ +on~${\Bbb C}^n$, $f=\sum c_\alpha z^\alpha$ normed by +\begin{equation} +\|f\|^2 = \sum \alpha! \, |c_\alpha|^2. +\label{eq:Fischer norm} +\end{equation} + +{\em Whenever\/ $\|\,{\cdot}\,\|$ appears in this paper, it designates this +norm.} We denote by~$\langle\,\, , \,\rangle$ the corresponding inner product +in~$F_n$. Finally, for $z$~and~$w$ in~${\Bbb C}^n$, $z\cdot w$~denotes +$\sum_1^n z_j w_j$. + +The main objective of this paper is to prove Theorem~\ref{theorem-Ellipsoid} +below. The special case $a_1=\dots=a_n=1$ (or, rather, a formulation equivalent +to this) is stated as Theorem~2 in \cite{Incorrect version}. Unfortunately, the +proof offered there is incorrect (the error, on p.~522, lies in applying +Lemma~1 to the series of polynomials $\sum h_m$: here $h_m$~is in~${\scr +P}_m$, but not homogeneous, so Lemma~1 is not applicable). Even more +unfortunately, the generalization of Theorem~2 of \cite{Incorrect version} +presented as the Corollary on p.~525 is also based on an invalid deduction. The +error here is the assertion that the analog of Theorem~3 for the space~$X_n$ +rather than $E_n$ can be proved by a similar argument---it cannot. Thus far we +do not know whether this Corollary is true as stated, or not. + +The (we hope) correct proof of Theorem~\ref{theorem-Ellipsoid} below (Theorem~2 +of \cite{Incorrect version}) is based on elementary potential theory, not using +techniques of \cite{Incorrect version} based on the Fischer +norm~(\ref{eq:Fischer norm}). Using the latter technique, we have not succeeded +in proving this theorem in full generality, but only within classes of entire +functions of limited growth (Theorem~\ref{theorem-Ellipsoid} below). But, in +return, we obtain an analogous result not only for the Laplace operator, but +for a fairly large class of differential operators. Thus, the Corollary on +p.~525 of \cite{Incorrect version} is shown to be true for each homogeneous +polynomial~$P$ of a certain `amenable' class (see Section~\ref{sec-Ellipsoid} +below), provided the given~$f$ is restricted to an appropriate subclass of the +entire functions. This is done in Theorem~\ref{theorem-Ellipsoid} of the +present paper. + +\section{Dirichlet's problem for the ellipsoid} +\label{sec-Ellipsoid} + +\begin{theorem} +\label{theorem-Ellipsoid} +Let\/ $\Omega=\{x\in{\Bbb R}^n : \sum_{j=1}^n a_j^{-1}x_j^2 < 1\}$, where +$a_j>0$. If $f$~is entire on\/ ${\Bbb C}^n$, the solution of the Dirichlet +problem +\begin{equation} +\vcenter{\openup\jot\ialign + {\strut\hfil$\displaystyle#$&$\displaystyle{}#$\hfil&\quad#\hfil\cr +\Delta u &= 0 &in\/ $\Omega$,\cr + u &= f &on $\partial\Omega$,\cr}} +\label{eq:Dirichlet} +\end{equation} +extends to a harmonic function on\/~${\Bbb R}^n$. {\rm (}Hence it extends to +an entire function on\/~${\Bbb C}^n$ satisfying\/ $\sum_1^n D_j^2 u = 0$, and +equal to~$f$ on the variety\/ $\{z\in{\Bbb C}^n : \sum_1^n a_j^{-2}z_j^2 = +1\}$.{\rm $\,$)} +\end{theorem} + +\begin{proof} +We can write the Taylor expansion of~$f$ as $f=\sum_0^\infty f_m$, where +$f_m\in{\scr H}_m$. The Dirichlet problem analogous to~(\ref{eq:Dirichlet}) +with $f_m$ in place of~$f$ has a unique solution $u_m\in{\scr P}_m$. (This +is well known, and is recalled for the reader's convenience in +Section~\ref{sec-Conclusion} below.) To complete the proof, we shall show that +$\sum_0^\infty u_m$ converges uniformly on compact subsets of ${\Bbb R}^n$. + +Let $\Gamma$ denote $\partial\Omega$. Let +\begin{equation} +u_m = u_{m,0}+u_{m,1}+\dots+u_{m,m} +\label{eq:u definition} +\end{equation} +denote the decomposition of $u_m$ into homogeneous polynomials; thus $u_{m,j}$ +is in~${\scr H}_j$ and harmonic. + +We shall now prove that, for every $R>0$, there is a constant $A(R)$ such that +\begin{equation} +\sum_{m=0}^\infty \sum_{k=0}^m |u_{m,k}(x)| \le A(R), +\quad \mbox{for $|x| \le R$}, +\label{eq:A(R) definition} +\end{equation} +which implies the desired convergence of~$\sum_0^\infty u_m$. +\end{proof} + +\begin{lemma} +\label{lemma-max limit} +Let $F_m=\max\{|f_m(x)| : x\in\Gamma\}$. Then $F_m^{1/m}\to 0$. +\end{lemma} + +\begin{proof} +The proof uses only that $\Gamma$~is a compact subset of~${\Bbb C}^n$, +contained in, say, the ball $B$:~$\{|z|\le\rho\}$. + +We have for $t\in{\Bbb C}$, +\[ +f(tz) = \sum_0^\infty t^m f_m(z). +\] +Fixing $z\in B$, $f_m(z)$~are the Taylor coefficients of the entire function +$t\mapsto f(tz)$ on~$\Bbb C$. By the Cauchy--Hadamard estimate, +\[ +|f_m(z)| \le \frac{\max \{|f(tz)| : t \le T\}}{T^m} +\] +holds for all $T>0$. Hence, +\[ +\max_{x \in B} |f_m(z)|\le\frac{\max\{|f(\zeta)|:|\zeta|\le\rho T\}}{T^m} +\] +Taking $m$th~roots and letting $m\to\infty$ gives +\[ +\limsup_{m\to\infty}\bigl(\max_{x\in B} |f_m(z)|\bigr)^{1/m} \le T^{-1} +\] +for arbitrary~$T$, implying the assertion. +\end{proof} + +\begin{remark} +The referee has remarked that it would be of interest to obtain a sharp form of +Lemma~\ref{lemma-max limit}, and has kindly supplied a proof that the +exponent~$n/2$ in~(\ref{eq:A(R) definition}) can be improved to $(n-2)/2$. +\end{remark} + +\begin{corollary*} +Let $v$, $v_k$ and $\Sigma$ be as in Lemma\/~{\rm \ref{lemma-max limit}}, and let +$D$~be a bounded open set in\/~${\Bbb R}^n$ containing the ball\/ +$\{|x|\le\rho\}$. Then, for $x\in\Sigma$, +\begin{equation} +|v_k(x)| \le {\scr C}_n k^{n/2} \rho^{-k} \cdot +\max_{x\in\partial D} |v(x)|, \quad k \ge 1. +\label{eq:bound on v} +\end{equation} +Also, $|v_0(x)| = |v_0(0)| \le \max\{|v(x)| : x\in\partial D\}$. +\end{corollary*} + +\begin{proof} +The statement concerning~$v_0$ is obvious, so suppose $k\ge1$, and without +loss of generality, assume $\max\{|v(x)| : x\in\partial D\}$ is~$1$. + +Then $|v(x)| \le 1$ for $|x|=\rho$, by the maximum principle, so $|v(\rho x)| +\le 1$ for $x\in\Sigma$. By the lemma, we have for $x\in\Sigma$, +\[ +|v_k(\rho x)| \le {\scr C}_n k^{n/2}, +\] +which gives (\ref{eq:bound on v}), since $v_k \in {\scr H}_k$. +\end{proof} + +\begin{proof}[of Theorem, completed] +We have, for $x\in\Gamma$, +\[ +|f_m(x)| \le \varepsilon_m^m, +\] +where $\varepsilon_m$ is a sequence which tends to~$0$. Hence, for +$x\in\Gamma$, $|u_m(x)|\le\varepsilon_m^m$. By the Corollary, the $u_{m,k}$ +in~(\ref{eq:u definition}) satisfy +\[ +|u_{m,k}(x)| \le {\scr C}_n k^{n/2} \rho^{-k} \varepsilon_m^m |x|^k, + \quad k \ge 1, +\] +and $|u_{m,0}| \le {\scr C}_n \varepsilon_m^m$, for all $x \in {\Bbb R}^n$, +where $\rho=\min_i a_i$. In particular, for $|x| \le R$, +\[ +|u_{m,k}(x)| \le {\scr C}'_n \cdot A^k\varepsilon_m^m R^k +\] +holds for every choice of +$A>\rho^{-1}$. Thus (\ref{eq:A(R) definition})~follows since $\sum_{m=0}^\infty +\varepsilon_m^m \sum_{k=0}^m (AR)^k$ is clearly convergent (for assuming, as we +may, $AR>1$, the inner sum is $\le C(AR)^{m+1}$, etc.). This completes the +proof of Theorem~\ref{theorem-Ellipsoid}. +\end{proof} + +\begin{remarks} +Variants of the theorem can easily be obtained from the above estimates, for +example, {\em if $f$~is of exponential type, so is~$u$}; indeed, $f$~is of +exponential type if and only~if $\max\{|f_m(z)| : z\in K\}$ does not exceed +$(A/m)^m$ (where $A=A(K)$~is some constant), holds for some (hence every) +compact~$K$ having $0$~as an interior point. The proof now follows in the +same way as before. +\end{remarks} + +\begin{definition} +Let $\Lambda$~denote the class of positive sequences $\{\lambda_m\}_0^\infty$ +with $\lambda_m \searrow 0$. To each sequence $\lambda=\{\lambda_m\}$ +in~$\Lambda$ we define $B_\lambda$ to be the set of all entire functions +$f = \sum_0^\infty f_m$ on~${\Bbb C}^n$ (where, as usual, $f_m \in {\scr +H}_m$) such that +\begin{equation} +\|f_m\| = o(\lambda_m^m)m^{m/2}, \quad m\to\infty, +\end{equation} +and +\begin{equation} +\|f\|_\lambda = \sup \lambda_m^{-m}m^{-m/2}\|f_m\|. +\label{eq:lambda definition} +\end{equation} +It is easy to check that $B_\lambda$ is a Banach space with the norm +$\|f\|_\lambda$. Moreover, it is separable, indeed $\sum_{m=0}^k f_m$ +converges to~$f$ as $k\to\infty$, for all $f \in B_\lambda$. +\end{definition} + +\begin{lemma} +If $g \in {\scr H}_m$, then +\[ +\sum_1^n \|D_j g\|^2 = m\|g\|^2. +\] +\end{lemma} + +\begin{proof} +$mg = \sum_1^n z_j D_j g$ by Euler's formula, so +\[ +m\|g\|^2 = \sum_1^n \langle z_jD_jg,g \rangle = \sum_1^n \|D_jg\|^2. +\] +\end{proof} + +\begin{notation} +Throughout this section, $Q$ denotes $\sum_1^n z_j^2$. +\end{notation} + +\begin{remark} +In terms of the Dirichlet problem, Theorem~\ref{theorem-Ellipsoid} says the +following. {\em For~$f\in B_\lambda$, where $\lambda$~satisfies (\ref{eq:lambda +definition})} (and hence, see the following remark, {\em for every entire~$f$ +of order~$<4$}), {\em the problem~(\ref{eq:lambda definition}), where\/ +$\Omega$~is the unit sphere, has a solution~$u$ that is (the restriction +to\/~$\Omega$ of) an entire function in~$B_\lambda$.} It would not be hard to +modify the proof to obtain an analogous result for ellipsoids rather than +spheres (and $B_\lambda$ replaced by some related class of entire functions), +by modifying Fischer's inner product so that $\sum_1^n D_j^2$ and +multiplication by $\sum_1^n a_j^{-2}z_j^2$ become adjoint operators. However, +we have been unable to obtain Theorem~\ref{theorem-Ellipsoid} (even for +spheres) by such methods. On the other hand (and this is the point of the +following section) these methods allow a generalization from $\sum_1^n D_j^2$ +to a large class of differential operators~$P(D)$ ($P$~being a homogeneous +polynomial). +\end{remark} + +\begin{remark} +To give some feeling for what (\ref{eq:lambda definition})~means, let us show: +{\em every entire function of order less than four is in~$B_\lambda$, for some +$\lambda\in\Lambda$ satisfying\/} (\ref{eq:lambda definition}). Indeed, suppose +$f$~is entire and +\begin{equation} +|f(z)| \le Ae^{|z|^\rho}, \quad z\in{\Bbb C}^n, +\end{equation} +where $A$ and $\rho$ are positive constants and $\rho<4$. +\end{remark} + +\section{Concluding remarks} +\label{sec-Conclusion} + +\subsection{} + +The basic question underlying this paper is that of finding global continuation +of the solution to Dirichlet's problem when such continuation is known both for +the equation of~$\partial\Omega$ and for the `data function'~$f$. Even when +extreme regularity is assumed, for example, $\partial\Omega$~algebraic and +$f$~entire, few results are known (even in two dimensions) about the maximal +domain to which the solution extends harmonically, let alone the nature of the +singularities that may arise. This is in contrast to the situation for Cauchy's +problem, where, for example, complete results are known in two dimensions, +based on the Schwarz function (compare \cite{Ref9,Ref12}). Moreover, G.~Johnson +\cite{Ref8} has obtained complete results for the Cauchy problem when the +initial data is an entire function restricted to a quadric surface (this for a +class of differential operators including the Laplacian). So far, there are no +results of this precision available for the Dirichlet problem. + +We have already spoken of the question of whether ellipsoids are characterized +by Theorem~\ref{theorem-Ellipsoid}. In this connection, recall that when +$\Omega$~is an ellipsoid, the solution of Dirichlet's problem with data +in~${\scr P}_m$ also lies in~${\scr P}_m$ (this is a classical result +from the study of ellipsoidal harmonics, and we used it in proving +Theorem~\ref{theorem-Ellipsoid}). This has a kind of converse, which one +readily sees as follows. + +\subsection{} + +Concerning the material in Sections +\ref{sec-Introduction}~and~\ref{sec-Ellipsoid}, some questions remain. +Especially, it seems of interest to know when the set of solutions +of~$P^*(D)(P-1)f=0$, $f\in B_\lambda$, not merely is finite-dimensional (for +which we gave sufficient conditions, in terms of $P$~and~$\lambda$) but +consists of $0$~alone. Perhaps the uniqueness assumption in +Theorem~\ref{theorem-Ellipsoid} could be omitted---we know of no +counterexample. + +\begin{acknowledgements} +This work was done while the first author was visiting Stockholm in the spring +of~1991. The first author is indebted to the Royal Institute of~Technology for +support and for providing a congenial research environment. +\end{acknowledgements} + +\begin{thebibliography}{99} + +\bibitem{Ref1} +{\bibname V. Bargmann}, `On a Hilbert space of analytic functions and an +associated integral transform', {\it Comm.\ Pure Appl.\ Math.}\ (1961) +187--214. +% +\bibitem{Ref2} +{\bibname F. A. Berezin}, `Covariant and contravariant symbols of operators', +{\it Math.\ USSR-Izv.}\ 6 (1972) 1117--1151. +% +\bibitem{Ref3} +{\bibname C. A. Berger \and L. A. Coburn}, `Toeplitz operators and quantum +mechanics', {\it J. Funct.\ Anal.}\ 68 (1986) 273--299. +% +\bibitem{Ref4} +{\bibname C. A. Berger \and L. A. Coburn}, `Toeplitz operators on the +Segal--Bargmann space', {\it Trans.\ Amer.\ Math.\ Soc.}\ 301 (1987) +813--829. +% +\bibitem{Ref5} +{\bibname V. Guillemin}, `Toeplitz operators in $n$~dimensions', {\it +Integral Equations Operator Theory\/} 7 (1984) 154--205. +% +\bibitem{Ref6} +{\bibname J. Janas}, `Toeplitz and Hankel operators on Bargmann spaces', +{\it Glasgow Math.\ J.} 30 (1988) 315--323. +% +\bibitem{Ref7} +{\bibname J. Janas}, `Unbounded Toeplitz operators in the Bargmann--Segal +space', {\it Studia Math.}, to appear. +% +\bibitem{Ref8} +{\bibname G. Johnson}, `The Cauchy problem in ${\Bbb C}^n$ for +second-order PDE with data on a quadric surface', in preparation. +% +\bibitem{Ref9} +{\bibname D. Khavinson \and H. S. Shapiro}, {\it The Schwarz potential in\/ +${\Bbb R}^n$ and Cauchy's problem for the Laplace equation}, +Research Report TRITA-MAT-1989-36 (Royal Institute of Technology, 1989). +% +\bibitem{Ref10} +{\bibname D. J. Newman \and H. S. Shapiro}, `A Hilbert space of entire +functions related to the operational calculus', mimeographed, Ann Arbor, 1964. +% +\bibitem{Ref11} +{\bibname D. J. Newman \and H. S. Shapiro}, {\it Fischer spaces of entire +functions}, Proc.\ Sympos.\ Pure Math II (Amer.\ Math.\ Soc., Providence, RI, 1968) 360--369. +% +\bibitem{Ref12} +{\bibname H. S. Shapiro}, {\it The Schwarz function and its generalization to +higher dimensions\/} (Wiley, 1991). +% +\bibitem{Incorrect version} +{\bibname H. S. Shapiro}, `An algebraic theorem of E.~Fischer, and the +holomorphic Goursat problem', {\it Bull.\ London Math.\ Soc.}\ 21 +(1989) 513--535. +% +\bibitem{Ref14} +{\bibname H. S. Shapiro}, `Analytic continuation of the solution to +Dirichlet's problem', in preparation. +% +\bibitem{Ref15} +{\bibname B. Yu.\ Sternin \and V. E. Shatalov}, `Continuation of solutions +of elliptic equations and localization of singularities', preprint, 1991. +\end{thebibliography} + +%% Note that there is no paragraph spacing between two affiliations +%% on one line + +\affiliationone{Department of Mathematics\\ +University of Arkansas\\ Fayetteville, AR 72701\\ USA} +% +\affiliationtwo{Mathematiska Institutionen\\ +Kungl.\ Tekniska H\"ogskolan\\ S-100 44 Stockholm\\ Sweden} + +\end{document} |