Difficult relationships between functional analysis and mathematical physics
type: Item
instance of: scholarly article
title: Difficult relationships between functional analysis and mathematical physics
zbMATH Open document ID: 0709.45004
author: Gaetano Fichera
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/244311
review text: In order to point out the basic role played by the choice of the norm for stating properties in problems of analysis and mathematical physics, the author develops and discusses three examples. Firstly the Volterra integral equation: \(U(x,t)=\exp (-x^ 2)+\int^{t}_{1}(2/\tau -6\tau^ 5x^ 2)U(x,\tau)d\tau,\) having as solution the function: \(u(t)\equiv U(x,t)=t^ 2\exp (-t^ 6x^ 2)\) defined for \(t\in [1,+\infty)\) with values in a Banach space S. The author comes to the conclusion that assuming as space S the space with the norm \(\| \nu \| =\int^{1}_{-1}| \nu (x)| dx\) the above solution u(t) is asymptotically stable. At the contrary the statement is not true if the space S has the norm \(\| \nu \| =\max_{[-1,1]}| \nu (x)|.\) A second example refers to the ``well-posedness'' of the Cauchy problem for the Laplace operator. The last example refers to the ``principle of fading memory'' in visco- elasticity, with consideration of the integral equation in \(\epsilon\) (t): \(\sigma (t)=G(0)\epsilon (t)+\int^{t}_{t_ 0}G(t-\tau)\epsilon (\tau)d\tau.\) The assumed functional space \(S_{\gamma}\), for \(\epsilon\) (t) is: \(\epsilon (t)\in L^ 2_{loc},\quad \int^{+\infty}_{0}| \epsilon (-s)|^ 2\exp (-2\gamma s)ds<+\infty,\) and the particular value considered for G(s) is \(1/2+1/2 \exp (-s).\) The author comes to the conclusion that for \(\gamma\leq 1/2\) the uniqueness theorem holds for the said integral equation (with \(t_ 0=-\infty)\). At the contrary for \(\gamma >1/2\) the theorem is not true.
Mathematics Subject Classification ID: 45D05
Mathematics Subject Classification ID: 45M10
Mathematics Subject Classification ID: 74D99
zbMATH DE Number: 4165685
zbMATH Keywords: Volterra integral equation
zbMATH Keywords: Banach space
zbMATH Keywords: Cauchy problem
zbMATH Keywords: Laplace operator
zbMATH Keywords: visco-elasticity
reviewed by: Silvio Nocilla
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4165685