On the majorization method for holomorphic solutions of linear partial differential equations
The differential equation \(\sum^{n}_{i=0}a_ i\partial^ nz(x,y)/\partial x^ i\partial y^{n-i}=f(x,y),\) where f(x,y) is holomorphic for \(| x| <r\), \(| y| <\rho\), possesses under the initial conditions \(\partial^ iz(0,y)/\partial x^ j=\partial^ jz(x,0)/\partial y^ j=0,\) \(i=0,...,k-1\) and \(j=0,...,n-k-1\) as well as certain conditions on the constant coefficients \(a_ i\) exactly one solution, which is holomorphic for \(| x| <pR/\alpha,\) \(| y| <(1-p)R\) with \(R=\min (\alpha r,\rho),\) where p is an arbitrary number with \(0<p<1\) and \(\alpha\) a number depending only on the coefficients \(a_ i\). This result generalizes partly a former theorem of the author [Z. Angew. Math. Mech. 60, 453-458 (1980; Zbl 0424.35002)]. It is sketched, how the result can be transformed to the case that homogeneous initial conditions are given on a curve through \(x=y=0\) being holomorphic at this point.
scientific article; zbMATH DE number 3886377