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authorNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
committerNorbert Preining <norbert@preining.info>2019-09-02 13:46:59 +0900
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+% 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
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+ \fi
+%
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+%
+%%% one (oldfss/noams)
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+ \def\Bbb{\protect\bf}
+ \def\scr{\protect\cal}
+ \fi
+
+%%% two (oldfss/amsone)
+ \iftwo
+ \input mssymb\relax % %%%MR own catcodes
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+ %
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+ \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}
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+ % set up 9pt Bbb for bibliography
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+ \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
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+ \def\bibliobbb{\textfont\msbfam=\ninemsb}
+ %
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+ \fi
+
+%%% four (nfssone/noams)
+ \iffour
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+ \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
+ %
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+ %
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+ \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}
+
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+
+%% 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}