×

zbMATH — the first resource for mathematics

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
PDF BibTeX XML Cite
Full Text: DOI