Parameter differentiation of linear operators with a parameter-dependent domain
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.
symmetric and regularly dissipative sesquilinear form
scientific article; zbMATH DE number 6106518