Divergence property of formal solutions for singular first order linear partial differential equations
Divergence property of formal solutions for singular first order linear partial differential equations
复制标题
奇异一阶线性偏微分方程形式解的发散性
DOI：
10.2977/prims/1195143361
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发表时间：
1999
影响因子：
1.2
通讯作者：
Masaki Hibino
中科院分区：
文献类型：
--
作者：
Masaki Hibino
本文研究了一类在原点处系数为全纯的一阶奇异线性偏微分方程的幂级数形式解的收敛性和散度：P(x, d)u(x) = al(x)Dlu(x)+b(x}u(x) = /(*), 1=1，且f (x)在原点处为全纯。这里的方程是奇异的，如果a， (0)=0 （y = 1，…）， < s f)。在这种情况下，已知在所谓的庞加莱条件下，如果{a，(x)}l=l产生一个简单理想，则每个形式解都是收敛的。然而，如果我们去掉这些条件，我们将看到，如果存在形式解，它可能是发散的。更准确地说，我们将通过由牛顿多面体确定的形式解的Gevrey阶来表征形式解的散度速率，牛顿多面体是牛顿多边形的一种推广，在具有不规则奇点的常微分方程的研究中很常见。§
This paper is concerned with the study of the convergence and the divergence of formal power series solutions of the following first order singular linear partial differential equation with holomorphic coefficients at the origin: d P(x,D)u(x) = al(x)Dlu(x)+b(x}u(x] = /(*), 1=1 with f ( x ) holomorphic at the origin. Here the equation is said to be singular if a , (0)=0 (y = 1, . . . , < s f ) . In this case, it is known that under the so-called Poincare condition, if {a,(x)}l=l generates a simple ideal, every formal solution is convergent. However if we remove these conditions, we shall see that the formal solution, if it exists, may be divergent. More precisely, we will characterize the rate of divergence of formal solutions via Gevrey order of formal solutions determined by a Newton Polyhedron, a generalization of Newton Polygon which is familiar in the study of ordinary differential equations with an irregular singular point. §