数理解析研究所講究録 1211 巻 2001 年 19-20

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Asymptotic Expansions for Solutions
to Semilinear Fuchsian Equations
Ingo Witt
University of Potsdam, Institute for Mathematics,PF601553, D-14415 Potsdam, Germany

E-mail address: ingo@math.uni-potsdam.de

Let $X$ be a$C^{\infty}$

manifold with boundary,(1)$\partial X$. We consider the semilinear elliptic equation$Au=F(x, B_{1}u, \ldots, B_{K}u)$on$X^{\mathrm{O}}=X\backslash \partial X$,
for $1\leq J\leq K$ ,for the unknown $u=u(x)$ , where. Hereis acertain)$\mu>\max_{1\leq J\leq K}\mu_{J}$ , and F\in C%(Xadmitting
functions on
is the space ofasymptotic type (see below) andthe data of which are prescribed by .
an asymptotic expansion as,ofmeans thatand in acollar neighborhood), takes the form
, $x\mapsto’(t, y)$ (so that
$A\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}$$\mathrm{x}\mathrm{R}^{K}$$B_{J}\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}_{\mathrm{F}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{s}}^{\mu J}(X)$$\mu \mathrm{F}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{s}(X)$$R$$(=C^{\infty}(\mathbb{R}^{K};C_{R}^{\infty}(X)))$$C^{\infty}$$C_{R}^{\infty}(X)$$X^{\mathrm{O}}$$R$$xarrow\partial X$$\mathcal{U}$$A\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}^{\mu}(X^{\mathrm{O}})$$A\in \mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}_{\mathrm{F}\mathrm{u}\mathrm{c}\mathrm{h}\mathrm{s}}^{\mu}(X)$$Y\approx\partial X$$\mathcal{U}arrow\approx[0,1)\cross Y$

(2)

$A$

$A=t^{-\mu} \sum_{j=0}^{\mu}a_{j}(t, y, D_{y})(t\partial_{t})^{j}$wherethen means that$a_{j}(t)=a_{j}(t, y, D_{y})\in C^{\infty}([0,1);\mathrm{D}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{f}^{\mu-j}(\mathrm{Y}))$$A$

is elliptic on$X^{\mathrm{o}}$$\partial X$,for $1\leq j\leq\mu$ . The ellipticity of$A$

and, in addition,$\sum_{j=0}^{\mu}\sigma^{\mu-j}(a_{j}(t))(y, \eta)(i\overline{\tau})^{j}\neq 0$

for all,$(t, y,\overline{\tau}, \eta)\in\tilde{T}^{*}([0,1)\mathrm{x}\mathrm{Y})\backslash \mathrm{O}$is the compressed
denotes the principal symbol of
where, and, being the covariable to .
cotangent bundle of $[0, 1)$$a_{j}(t),\overline{T}^{*}([0,1)\mathrm{x}Y)$$\sigma^{\mu-j}(a_{j}(t))$$\cross \mathrm{Y}$$\overline{\tau}=t\tau$$t$$\tau$

Atypical result reads as follows:

be elliptic in the above sense. Further let be a solution to (1) inTheorem 1. Let
the class of extendible distributions such that $Bju\in L^{\infty}(X)$ for all $1\leq J\leq K$ . Then,
under some natural technical assumptions, there exists an asymptotic type $P$ such that, $R$ , and the$u\in C_{P}^{\infty}(X)$ .
Furthermore, $P$ can be expressed in terms of,,.
sense) of close toconormal order ( $=order$ of flatness, in an$A$$u$$A_{f}B_{1}$$L^{2}$

Asymptotic expansions for solutions $u=u(x)$ to (1) are of the form$u(x) \sim\sum_{j=0}^{\infty}\sum_{k=0}^{m_{j}}t^{-p_{j}}\log^{k}tc_{jk}(y)$as$tarrow+\mathrm{O}$$\ldots$$\partial X$$u$,$B_{K}$

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where$p_{j}\in \mathbb{C}$,${\rm Re} p_{j}arrow-\infty$

as $jarrow\infty$ ,$m_{j}\in \mathrm{N}$, and the coefficients $c_{jk}(y)$ vary in a

finite-dimensional subspace. In the notion of asymptotic type introduced$[3,4]$,by B.-W. Schulze
the sequenceis regarded as constituting an$L_{j}\subset C^{\infty}(\mathrm{Y})$$\{(p_{j}, m_{j}, L_{j})\}_{j=0}^{\infty}$

asymptotic type.

Results of the sort of Theorem 1rely on arefined notion of asymptotic type in which
linear relations between the various coefficients, even for different , are allowed.
Whereas in the former notion of asymptotic tyPe only the aspect of the “production” ofasymptotics is emphasized, now also the aspect of their “annihilation” is underscored.
The basic technical question to answer is what kind of linear relation between the coefficientsis admissible $[2, 6]$ . The resulting notion of asymptotic type both turns out tobe coordinate-invariant [2] and seems to admit adequate pseudodifferential calculi $[1, 5]$ .
These results were jointly obtained with Liu Xiaochun, Wuhan University, China.
$c_{jk}\in L_{j}$$j$$c_{jk}$REFERENCES[1] X. Liu, A cone pseudodifferential calculus on the half-line with respect to conormal asymptotics of$a$given type. Ph.D. thesis, Department of Mathematics, Wuhan University, Wuhan, March 2000.
[2] X. Liu and I. Witt, Asymptotic expansions for bounded solutions to semilinear Fuchsian equations. In
preparation.
[3] S. Rempel and B.-W. Schulze, Asymptotics for elliptic mixed boundary problems. Math. Research,
vol. 50, Akademie Verlag, Berlin, 1989.
[4] B.-W. Schulze, Boundary value problems and singular pseudO-differential operators. Wiley Ser. Pure
Appl. Math., J. Wiley, Chichester, 1998.
[5] I. Witt, Explicit algebras with the Leibniz-Mellin translation product. Preprint 99/2, Institute ofMathematics, University of Potsdam, Potsdam, 1999.
[6] I. Witt, On the factorization of meromorphic Mellin symbols. Preprint 99/5, Institute of Mathematics,
University of Potsdam, Potsdam, 1999.