高阶全纯偏微分方程的解析芽形式Gevrey解
Formal Gevrey solutions -- in analytic germs -- for higher order holomorphic PDEs
摘要：
本文考虑一类全纯偏微分方程，其奇异轨迹由解析映射$P$的零点集给出，其中$P$具有$P(0)=0$。我们的目标是建立形式幂级数级数解的存在唯一性条件，并确定其发散率。事实上，我们证明，解决方案是Gevrey在$P$，发散性提供新的信息，而相比，经典的Gevrey类。如果$P$在$0$处不是奇异的，我们还提供了Poincaré条件来恢复收敛解.我们的策略是扩展的尺寸和升降机给定的偏微分方程的奇异偏微分方程的结果可以应用的问题。最后，例子中的Gevrey类$P$是最佳的。
英文摘要：
We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map $P$ with $P(0)=0$. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to determine their divergence rate. In fact, we prove that the solution is Gevrey in $P$, giving new information on divergency while compared to the classical Gevrey classes. If $P$ is not singular at $0$, we also provide Poincaré conditions to recover convergent solutions. Our strategy is to extend the dimension and lift the given PDE to a problem where results of singular PDEs can be applied. Finally, examples where the Gevrey class in $P$ is optimal are included.
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