A necessary condition for \(H^ \infty\)-well-posed Cauchy problem of Schrödinger type equations with variable coefficients
type: Item
instance of: scholarly article
title: A necessary condition for \(H^ \infty\)-well-posed Cauchy problem of Schrödinger type equations with variable coefficients
zbMATH Open document ID: 0794.35113
author: Shigeomi Hara
published in: Journal of Mathematics of Kyoto University
publication date: 1993-04-19T00:00:00Z
review text: On considère le problème de Cauchy \[ Lu=f,\;u|_{t=0}=u_ 0,\quad 0\leq t\leq T,\quad x\in\mathbb{R}^ n, \tag{1} \] où \[ L={1 \over i} \partial_ t-{1 \over 2} \sum_{1 \leq j,k \leq n} \partial_{x_ j} \left( g^{jk} (x) \partial_{x_ k} \right)+\sum_{1 \leq j \leq n} b^ j(x) \partial_{x_ j}+c(x), \] est uniformément elliptique, \(g^{jk}\), \(b^ j\), \(c\in B^ \infty (\mathbb{R}^ n)\), \(u,f \in {\mathcal E}^ 0_ t ([0,T]; H^ \infty(\mathbb{R}^ n))\) et \(u_ 0\in H^ \infty (\mathbb{R}^ n)\). On trouve une condition (portant sur \(b^ j\) et le flot hamiltonien de \(H(x,p)={1\over 2}\sum_{j,k}g^{jk}(x)p_ jp_ k)\) nécessaire afin que le problème (1) admette une unique solution, généralisant celle de \textit{W. Ichinose} [Osaka J. Math. 24, 853-886 (1987; Zbl 0654.35042)], du cas \(L^ 2\). On applique aussi les idées de Ichinose: estimation a priori par le théorème du graphe fermé et construction d'une solution asymptotique.
Mathematics Subject Classification ID: 35Q40
Mathematics Subject Classification ID: 37C10
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 109384
zbMATH Keywords: condition for a unique solution
zbMATH Keywords: elliptic equation
reviewed by: Viorel Iftimie
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/kjm/1250519537
OpenAlex ID: W1498311804
DOI: 10.1215/KJM/1250519537
description: scientific article; zbMATH DE number 109384