Lions' representation theorem and applications

Wolfgang Arendt (1) , Isabelle Chalendar (2) , Robert Eymard (2)

1 Universität Ulm - Ulm University [Ulm, Allemagne]
2 LAMA - Laboratoire Analyse et de Mathématiques Appliquées

The Lions' Representation Theorem (LRT) is a version of the Lax-Milgram Theorem where completeness of one of the spaces is not needed. In this paper, LRT is deduced from an operator-theoretical result on normed spaces, which is of independent interest. As an example, we give a new characterization of dissipativity. The main part of the paper is a theory of derivations, based on LRT, which we develop. Its aim is to establish well-posedness results, not only for evolution in time but also for more general settings in terms of this new notion of derivation. One application concerns non-autonomous evolution equations with a new kind of boundary condition where values at the initial and final time are mixed.

Mots clés: Lions' representation theorem; non-autonomous evolution equations; boundary conditions; dissipative operators; Lions' representation theorem

Domaines: Analyse numérique [math.NA]

Wolfgang Arendt, Isabelle Chalendar, Robert Eymard. Lions' representation theorem and applications. Journal of Mathematical Analysis and Applications, 2023, 522 (2), pp.126946. ⟨10.1016/j.jmaa.2022.126946⟩. ⟨hal-03903110⟩