The reducibility of the boundary conditions in the one-parameter family of elliptic linear boundary value problems. II
type: Item
instance of: scholarly article
title: The reducibility of the boundary conditions in the one-parameter family of elliptic linear boundary value problems. II
zbMATH Open document ID: 0705.35008
author: Ryuichi Ashino
published in: Osaka Journal of Mathematics
publication date: 1989-01-01T00:00:00Z
review text: [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.
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 42B10
zbMATH DE Number: 4157014
zbMATH Keywords: unilateral boundary value problems
zbMATH Keywords: reducibility
zbMATH Keywords: Cauchy problems
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4157014