On the uniqueness theorem for generalized solutions of initial-boundary problems for the Maguerre-Vlasov vibrations of shallow shells with clamped boundary conditions
type: Item
instance of: scholarly article
title: On the uniqueness theorem for generalized solutions of initial-boundary problems for the Maguerre-Vlasov vibrations of shallow shells with clamped boundary conditions
zbMATH Open document ID: 0936.35116
author: V. I. Sedenko
published in: Applied Mathematics and Optimization
publication date: 2000-05-18T00:00:00Z
review text: The author studies the problem of uniqueness of (weak) solutions of a nonlinear system of evolution equations describing the deformations of elastic shallow shells in the variables \(u\), \(v\) and \(w\). The first two describe the longitudinal point displacements on the middle surface of the shell, and \(w\) describes the tranverse displacements. Such a system is known in the literature as the complete Marguerre-Vlasov nonlinear equations of shallow shells. Global existence of (weak) solutions is already known. Uniqueness is much harder. Essentially because of the nonlinearity it is a critical case taking into account available embedding techniques. In this paper the author uses a new technique (due to him) to show uniqueness. The method relies on the energy inequality in negative spaces and on estimates of the type \[ \max_{x\in \Omega}\bigl |(T_Nh)(x) \bigr |\leq C(\log N)^{1/2}\|h\|_{H^1} \] and \(\|(I-T_N)h\|_{L^2} \leq N^{-1} \|h\|_{H^2}\) for a suitable sequence of operators \(T_N\), \(N=2, 3, \dots,\) where \(B\) is a positive constant (independent of \(N)\). This paper is a good contribution to the subject.
reviewed by: G. Perla Menzala
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 74K25
zbMATH DE Number: 1289880
zbMATH Keywords: smoothing operators
zbMATH Keywords: a new technique to show uniqueness
zbMATH Keywords: Marguerre-Vlasov nonlinear equations
MaRDI profile type: Publication
full work available at URL: s002459900108
OpenAlex ID: W2079643809
DOI: 10.1007/S002459900108
description: scientific article; zbMATH DE number 1289880