On continuity of functions with values in various Banach spaces

Walter Strauss

Vol. 19 (1966), No. 3, 543–551

DOI: 10.2140/pjm.1966.19.543

Abstract

In the discussion of weak solutions of certain kinds of partial differential equations, a crucial point, which is isolated in this paper, concerns the proof of identities of energy type and of the continuity of the solutions, which two questions are intimately related. The continuity referred to is with respect to a distinguished independent real variable t, the other variables being suppressed into some Banach space.

In §2 a simple argument shows that an essentially bounded function of t with values in a space V is automatically weakly continuous in V provided it is weakly continuous in some larger space.

In §3 conditions are found under which a square-integrable function u(t) with values in V is strongly continuous in V (Theorem 3.3). Roughly speaking, the main condition is that there exist self-adjoint linear operators A(t) coercive with respect to V such that A(⋅)u(⋅) and du∕dt lie in spaces which are dual to each other.

Mathematical Subject Classification

Primary: 47.90

Secondary: 34.95

Milestones

Received: 24 September 1965

Published: 1 December 1966