Parameter differentiation of linear operators with a parameter-dependent domain
type: Item
instance of: scholarly article
title: Parameter differentiation of linear operators with a parameter-dependent domain
zbMATH Open document ID: 1261.46035
DOI: 10.1134/S1064562412040436
author: F. E. Lomovtsev
published in: Doklady Mathematics
publication date: 2012-11-15T00:00:00Z
review text: Adapted from the introduction: The proof of the Hadamard well-posedness of mixed problems for nonstationary differential equations brings about the need for the time differentiation of operators generated by differential operators and spatial boundary conditions with time-dependent coefficients. If the conditions in the boundary conditions depend on time, this differentiation is undetermined. For such operators, the concept of weak time derivative is introduced which is sufficient for the well-posedness of the nonstationary mixed problems. Two formulas are derived for the weak time derivatives of operators defined by sesquilinear forms and operator forms. Methods for verifying the applicability conditions and for calculating weak derivatives according to these formulas are elaborated. Symmetric and regularly dissipative sesquilinear forms were already studied by \textit{K. Friedrichs} [Math. Ann. 109, 465--487 (1934); correction ibid. 110, 777--779 (1935; Zbl 0008.39203, JFM 60.1078.01)] and \textit{T. Kato} [J. Math. Soc. Japan 13, 246--274 (1961; Zbl 0113.10005); ibid. 14, 242--248 (1962; Zbl 0108.11203)]. Lions also used sesquilinear forms to define linear operators and boundary value problems [\textit{J. L. Lions}, Équations différentielles opérationnelles et problèmes aux limites. Berlin-Göttingen-Heidelberg: Springer-Verlag (1961; Zbl 0098.31101)]. In contrast to these and other works, in this paper, sesquilinear forms on the product of different Hilbert spaces are used.
reviewed by: Ludwig Kohaupt
Mathematics Subject Classification ID: 46G05
Mathematics Subject Classification ID: 47A07
Mathematics Subject Classification ID: 35M10
Mathematics Subject Classification ID: 35A99
zbMATH DE Number: 6106518
zbMATH Keywords: parameter differentiation
zbMATH Keywords: Hadamard well-posedness
zbMATH Keywords: nonstationary differential equation
zbMATH Keywords: weak time derivative
zbMATH Keywords: symmetric and regularly dissipative sesquilinear form
MaRDI profile type: Publication
cites work: Spektraltheorie halbbeschränkter Operatoren und Anwendung auf die Spektralzerlegung von Differentialoperatoren. I
cites work: Fractional powers of dissipative operators
cites work: A generalization of Lions' theory to first-order evolution differential equations with smooth operator coefficients. I
description: scientific article; zbMATH DE number 6106518