## Journal of the Mathematical Society of Japan

- J. Math. Soc. Japan
- Volume 52, Number 4 (2000), 767-792.

### Growth property and slowly increasing behaviour of singular solutions of linear partial differential equations in the complex domain

#### Abstract

Consider a linear partial differential equation in ${C}^{d+1}P(z,\partial )u\left(z\right)=f\left(z\right)$, where $u\left(z\right)$ and $f\left(z\right)$ admit singularities on the surface $\{{z}_{0}=0\}$. We assume that $\left|f\right(z\left)\right|\le A|{z}_{0}{|}^{c}$ in some sectorial region with respect to ${z}_{0}$. We can give an exponent ${\gamma}^{*}>0$ for each operator $P(z,\partial )$ and show for those satisfying some conditions that if $\forall \u03f5>0\exists {C}_{\u03f5}$ such that $\left|u\right(z\left)\right|\le {C}_{\u03f5}\mathrm{exp}\left(\u03f5\right|{z}_{0}{|}^{-{\gamma}^{*}})$ in the sectorial region, then $\left|u\right(z\left)\right|\le C|{z}_{0}{|}^{{c}^{\prime}}$ for some constants ${c}^{\prime}$ and $C$.

#### Article information

**Source**

J. Math. Soc. Japan, Volume 52, Number 4 (2000), 767-792.

**Dates**

First available in Project Euclid: 10 June 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jmsj/1213107109

**Digital Object Identifier**

doi:10.2969/jmsj/05240767

**Mathematical Reviews number (MathSciNet)**

MR1775389

**Zentralblatt MATH identifier**

0966.35006

**Subjects**

Primary: 35A20: Analytic methods, singularities

Secondary: 35B05: Oscillation, zeros of solutions, mean value theorems, etc. 35B40: Asymptotic behavior of solutions

**Keywords**

Singular solutions complex partial differential equations

#### Citation

ŌUCHI, Sunao. Growth property and slowly increasing behaviour of singular solutions of linear partial differential equations in the complex domain. J. Math. Soc. Japan 52 (2000), no. 4, 767--792. doi:10.2969/jmsj/05240767. https://projecteuclid.org/euclid.jmsj/1213107109