Singular Solutions with Asymptotic Expansion of Linear Partial Differential Equations in the Complex Domain
Sunao Ōuchi · Publications of the Research Institute for Mathematical Sciences · 1998
We consider a linear partial differential equation with holomorphic coefficients in a neighbourhood of z=0 in C^{d+1} , P(z, ∂) u(z) = f(z), where u(z) and f(z) admit singularities on the surface K=\{z_0=0\} . Our main result is the following: For the operator P we define an exponent γ^* called the minimal irregularity of K and show that if u(z) grows at most exponentially with exponent γ^* as z_0 tends to 0 and if f(z) has a Gevrey type expansion of exponent γ^* with respect to z_0 , then u(z) also has the same one.