A Sharp Existence and Uniqueness Theorem for Linear Fuchsian Partial Differential Equations (Asymptotic Analysis and Microlocal Analysis of PDE)
Jose Ernie C. Lope · Institutional Repositories DataBase (IRDB) · 2001
This paper considers the equation $Pu=f$ , where $u$ and $f$ are contin- uous with respect to $t$ and holomorphic with respect to $z$ , and $\mathcal{P}$ is the linear Fuchsian partial differential operator $P$ $=(tD_{t})^{m}+ \sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}aj,\alpha(t, z)(\mu(t)D_{z})^{\alpha}(tD_{t})^{j}$ .We will give asharp form of unique solvability in the following sense: we can find adomain $\Omega$ such that if $f$ is defined on $\Omega$ , then we can find a unique solution $u$ also defined on Q.1Introduction and Result Denote by $\mathrm{N}$ the set of nonnegative integers, and let $(t, z)=(t, z_{1}, \ldots, z_{n})\in$ $\mathbb{R}\cross \mathbb{C}^{n}$ .Let $R>0$ be sufficiently small, and for $\rho\in(0, R]$ , let $B_{\rho}$ be the polydisk { $z\in \mathbb{C}^{n}$ ; $|z:|0$ .Then we denote by $C^{0}([0, T], A(D))$ the set of functions continuous on the interval $[0, T]$ and valued in the space $A(D)$ .We say that acontinuous, positive-valued function $\mu(t)$ on the interval $(0, T)$ is aweight function if $\mu(t)$ is increasing and the function $\varphi(t)=\int_{0}^{t}\frac{\mu(s)}{s}ds$ (1.1) is well-defined on (0, T), i.e., the integral on the right is finite.(See Tahara [7].)Consider now the linear partial differential operator $\mathcal{P}=(tD_{t})^{m}+\sum_{j=0}^{m-1}\sum_{|\alpha|\leq m-j}a_{j,\alpha}(t, z)(\mu(t)D_{z})^{\alpha}(tD_{t})^{j}$ .(1.2) Here, $D_{t}=\partial/\partial t$ and $D_{z}=$ $(\partial/\partial z_{1},$ \ldots , $\partial/\partial z_{n});\mu(t)$ is aweight function; and the coefficients $a_{j,\alpha}(t,$ z) belong in the space $C^{0}([0,$ T], $A(B_{R}))$ , i.e., for any