The reducibility of the boundary conditions in the one-parameter family of elliptic linear boundary value problems. II
[For part I see the preceding review.] Let \(P_ 1(D)\) and \(P_ 2(D)\) be linear partial differential operators with constant coefficient and the order of \(P_ 1\) with respect to \(x_ 1\) be m, that of \(P_ 2\) be \(m'\), and \(m>m'\). Consider the unilateral boundary value problems: \[ (\epsilon^{m-m'}P_ 1(D)+P_ 2(D))u(x)=0\text{ in } (0,\delta)\times R^{n-1};\quad b_{j_ k}(D)u(x)|_{x_ 1\downarrow 0}=\phi_ k(x'),\quad k=1,...,\mu. \] Here \(\Phi =(\phi_ 1,...,\phi_{\mu})\) belongs to \({\mathcal F}^{- 1}(C^{\infty}_ 0(R^{n-1}))^{\mu}\), where \({\mathcal F}^{-1}\) denotes the inverse Fourier transformation, and \(\mu\) must be chosen so that the bounded solutions are uniquely determined. We have the similar results as those on the reducibility of the family of elliptic boundary value problems. Next, consider the singulary perturbed Cauchy problems for Kowalewskian operators of order m. If the Cauchy problems are uniquely solvable and the limit \(u_ 0\) of the solutions \(u_{\epsilon}\) exists in C(R;\({\mathcal D}'(R^{n-1}))\), then \(u_ 0\) satisfies the first \(m'\) initial conditions.
scientific article; zbMATH DE number 4157014