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

Abstract

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