The behaviour near the characteristic surface of singular solutions of linear partial differential equations in the complex domain
Let \(L(z,\partial_ z)\) be a linear partial differential operator with holomorphic coefficients in a domain \(\Omega \subset {\mathbb{C}}^{n+1}\). Let K be a nonsingular complex hypersurface in \(\Omega\) and characteristic for \(L(z,\partial_ z)\). The following problem is considered for the equation \(L(z,\partial_ z)u(z)=f(z)\), where u(z) and f(z) are holomorphic except on K: when we know the behaviour of f(z) near K and the growth order of u(z) near K, what can we say about the behaviour of u(z)? It is one of results that under some conditions on \(L(z,\partial_ z)\), if u(z) has some growth order and f(z) has some Gevrey type asymptotic expansion in a sector S with edge K, then u(z) has the same type expansion in S. It is also considered the case when f(z) behaves like a pole or logarithmically, and another result is that u(z) also behaves in the similar way to f(z) under some conditions. The details will be published elsewhere.
scientific article; zbMATH DE number 4159175
scientific article