On Continuation of Gevrey Class Solutions of Linear Partial Differential Equations

Abstract

Dedicated to Professor Hikosaburo KOMATSU for his 60-th anniversary Abstract. We give a sufficient condition for the removability of thin singularities of Gevrey class solutions of linear partial differential equations. In §1we give a sufficient condition for the removability in the case of equations with constant coefficients. Then in §2 we discuss the necessity of the condition and construct non-trivial solutions with irremovable thin singularities for some class of equations. In §3 we give a sufficient condition for the removability of thin singularities of Gevrey class solutions in the case of equations with real analytic coefficients. In this article, we gather results on continuation to thin singularity (or removability of thin singularities) of Gevrey class solutions to linear par- tial differential equations. Some of the results given here are easily derived from Grushin's pioneering works on continuation of C ∞ solutions and from the author's former works on continuation of regular solutions. But it will be worth gathering them all to an article, because they may not be ob- vious for the readers who are not specialized in this subject. Moreover it will be adequate to dedicate this to Professor Hikosaburo Komatsu, who devoted his half carreer to the study of ultra-differentiable functions and ultradistributions. Here is a brief plan of the present article. The first two sections treat equations with constant coefficients. In §1we give a sufficient condition for