Singularities of solutions of equations with noninvolutive characteristics. I: The case of second order Fuchsian equations
Let \(L_ i(t,z,\partial_ t, \partial_ z)\) (\(i=1,2\)), \((t,z)\in\mathbb{C}^ 1\times \mathbb{C}^ n\), be linear partial differential operators with the order \(i\): \[ \begin{aligned} L_ 2(t,z,\partial_ t,\partial_ z) &= \partial^ 2_ t+A_ 1 (t,z,\partial_ z) \partial_ t+ A_ 2(t,z,\partial_ z),\\ L_ 1(t,z,\partial_ t, \partial_ z) &= a_ 0(t,z)\partial_ t+ \sum^ n_{i=1} a_ i(t,z)\partial_{z_ i}+ d(t,z), \end{aligned} \] where their coefficients are holomorphic in a neighbourhood \(\Omega\) of the origin. Let \(K_ i\) (\(i=1,2\)) be the characteristic surfaces of \(L_ 2(t,z,\partial_ t,\partial_ z)\) through \(t=z_ 1=0\). Put \[ L(t,z,\partial_ t, \partial_ z)= t^ 2 L_ 2(t,z,\partial_ t, \partial_ z)+ tL_ 1(t,z,\partial_ t, \partial_ z)+c(t,z), \] which is a Fuchsian operator with respect to \(K_ 0=\{t=0\}\). Consider the following equation \(L(t,z,\partial_ t, \partial_ z)=u(t,z)=f(t,z)\), where \(f(t,z)\) is holomorphic on \(\Omega-K_ 1\) and \(u(t,z)\) is holomorphic in a neighbourhood of \((\Omega-K_ 1)\cap \{t=0\}\). In this paper an integral representation of \(u(t,z)\) is obtained under some conditions on the characteristic exponent, and it is shown that \(u(t,z)\) is holomorphic on the universal covering space of \(\Omega'- \bigcup^ 2_{i=0} K_ i\) for a small neighbourhood \(\Omega'\) of the origin. The following Fuchsian operator is also treated: \(L(t,z,\partial_ t, \partial_ z)= tL_ 2(t,z,\partial_ t, \partial_ z)+ L_ 1(t,z,\partial_ t, \partial_ z)\), for which Cauchy problem \(L(t,z,\partial_ t, \partial_ z) u(t,z)=f(t,z)\), \(u(0,z)=u_ 0(z)\), is considered and similar results to the preceding are obtained.
scientific article; zbMATH DE number 469059