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Convexity Properties for Weak Solutions of Some Differential Equations in Hilbert Spaces

Published online by Cambridge University Press:  20 November 2018

S. Zaidman
S. Zaidman*
Affiliation:
University of Montreal
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In this work we obtain a simultaneous extension of Theorems 1.6 and 1.7 in Agmon and Nirenberg (1), together with a partial extension of the result on backward unicity for parabolic equations by Lions and Malgrange (4).

Let H be a Hilbert space. (·) and | | are the notations for the scalar product and the norm in this space. Consider in H a family B(t), 0 ≤ t ≤ T, of closed linear operators with dense domain DB(t) (varying) with t. Let L2(0, T, H) be the space of Bochner square-integrable vector-valued functions with values in H. Our main result is the following

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 17 , 1965 , pp. 802 - 807
Copyright
Copyright © Canadian Mathematical Society 1965

References

1. Agmon, S. and Nirenberg, L., Properties of solutions of ordinary differential equations in Banach spaces, Comm. Pure Appl. Math., 16 (1963), 121–239.Google Scholar
2. Krein, S. G., On some classes of correctly posed boundary value problems, Dokl. Akad. Nauk., SSSR, N.S., 114 (1957), 1162–1165.Google Scholar
3. Lions, J. L., Equations différentielles opérationnelles et problèmes aux limites (Berlin, 1961).Google Scholar
4. Lions, J. L. and Malgrange, B., Sur Vunicité rétrograde dans les problèmes mixtes paraboliques, Math. Scand., 8 (1960), 277–286.Google Scholar