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Borel summability of formal solutions of some first order singular partial differential equations and normal forms of vector fields. (English) Zbl 1082.35046
For a holomorphic vector field $$L=\sum^d_{l=1}X_l(z)\partial_{z_l}$$ degenerating at $$z=0$$, the author considers the equation $$Lu=F(z,u)$$ and studies the Borel summability of a formal power series solution in the case where Jacobi matrix $$((\partial X_i/\partial_{z_j}(0))$$ has zero eigenvalues. Assume that the set $$\{z\in\Omega_0$$; $$X_j(0)=0$$, $$j=1, \dots,d\}$$ is a complex submanifold with codimension $$d_1$$, and Jacobi matrix $$((\partial X_i/ \partial_{Z_j}(0))$$ has rank $$d_0$$ and the eigenvalues $$\lambda_1,\dots, \lambda_{d_0}$$, and their convex hull does not contain the origin. Moreover assume $$d_1=d_0+1$$ and that there exist $$\varphi(z)$$, $$\rho(z)$$ such that $$L \varphi=\rho\varphi^\sigma$$, $$\rho(0)\neq 0$$, where $$\sigma$$ means the multiplicity of $$L$$. Then the author proves that if for all $$m=(m_1, \dots, m_{d_0})\in\mathbb{N}^{d_0}$$, $$\sum^{d_0}_{i=1}m_i\lambda_i-\frac{\partial F} {\partial u}(0,0)\neq 0$$ holds, there exists a unique formal solution $$\widetilde u(z)$$ of the above equation and moreover there is a holomorphic local coordinates system $$(x(z),y(z),t(z))\in \mathbb{C}^{d_0}\times\mathbb{C}^{d-d_0-1}\times \mathbb{C}$$ in a neighborhood $$\Omega$$ of the origin such that $$(x(0),y(0),t(0))=0$$, $$\Sigma\cap \Omega=\{x_1(0)=\cdots=x_{d_0}(0)=t(0)=0\}$$ and $$u(x,y,t)$$ which is a genuine solution of the above equation satisfying $$\widetilde u(z)= u(x(z),y(z), t(z))$$ and $\left| u(x,y,t)-\sum^{n-1}_{n=0}u_n(x,y) t^n\right|\leq AB^N \left(\Gamma \biggl(\frac{N} {\gamma}+1\biggr)\right) |t|^N$ for all $$N\in\mathbb{N}$$ in $$\{(x,y)$$; $$|x|<r,|y|,r\} \times\{0<|t|<r_0, |\arg t\theta|<\pi/2\gamma+\delta\}$$ with some positive constants $$A,B,r,r_0,\theta,\gamma,\delta$$.

##### MSC:
 35C10 Series solutions to PDEs 35C20 Asymptotic expansions of solutions to PDEs 35A20 Analyticity in context of PDEs 35F20 Nonlinear first-order PDEs 35F05 Linear first-order PDEs
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